Characterizing Curvatures and Stresses in Thin-Film Structures on Substrates having Spatially Non-Uniform Variations

ABSTRACT

Techniques and devices are described to use spatially-varying curvature information of a layered structure to determine stresses at each location with non-local contributions from other locations of the structure. For example, a local contribution to stresses at a selected location on a layered structure formed on a substrate is determined from curvature changes at the selected location and a non-local contribution to the stresses at the selected location is also determined from curvature changes at all locations across the layered structure. Next, the local contribution and the non-local contribution are combined to determine the total stresses at the selected location.

This application claims the benefits of the following three U.S.Provisional Applications: 1. No. 60/748,338 entitled “EFFECT OF THINFILM/SUBSTRATE RADII ON STONEY FORMULA FOR THIN/SUBSTRATE SUBJECTED TONON-UNIFORM” and filed on Dec. 6, 2005; 2. No. 60/812,339 entitled “ONTHE STONEY FORMULA FOR NON-UNIFORM FILM THICKNESS IN A THINFILM/SUBSTRATE SYSTEM SUBJECT TO NON-UNIFORM MISFIT STRAIN” and filed onJun. 8, 2006; and 3. No. 60/832,496 entitled “ON THE STONEY FORMULA FORA THIN FILM/SUBSTRATE SYSTEM WITH NON-UNIFORM SUBSTRATE THICKNESS” andfiled on Jul. 21, 2006.

This application is also a continuation-in-part application of andclaims the benefits of co-pending U.S. patent application Ser. No.11/432,663 (attorney docket No. 06618-956001) entitled “TECHNIQUES ANDDEVICES FOR CHARACTERIZING SPATIALLY NON-UNIFORM CURVATURES AND STRESSESIN THIN-FILM STRUCTURES ON SUBSTRATES WITH NON-LOCAL EFFECTS” and filedon May 10, 2006. The patent application Ser. No. 11/432,663 claims thebenefits of the following two U.S. Provisional Applications: 1. No.60/679,328 entitled “NON-UNIFORM, AXISYMMETRIC MISFIT STRAIN IN THINFILMS BONDED ON PLATE SUBSTRATES/SUBSTRATE SYSTEMS: THE RELATION BETWEENNON-UNIFORM FILM STRESSES AND SYSTEM CURVATURES” and filed on May 10,2005; and 2. No. 60/723,302 entitled “SPATIALLY NON-UNIFORM, ISOTROPICMISFIT STRAIN IN THIN FILMS BONDED ON PLATE SUBSTRATES: THE RELATIONBETWEEN NON-UNIFORM FILM STRESSES AND SYSTEM CURVATURES” and filed onOct. 4, 2005. The patent application Ser. No. 11/432,663 is also acontinuation-in-part application of and claims the benefit of U.S.patent application Ser. No. 11/080,087 entitled “TECHNIQUES FORANALYZING NON-UNIFORM CURVATURES AND STRESSES IN THIN-FILM STRUCTURES ONSUBSTRATES WITH NON-LOCAL EFFECTS” and filed on Mar. 14, 2005, publishedas US 2005-0278126 A1 and PCT/US2005/009406.

In addition, this application is a continuation-in-part application ofand claims the benefits of U.S. patent application Ser. No. 11/080,087(attorney docket No. 06618-941001) entitled “TECHNIQUES FOR ANALYZINGNON-UNIFORM CURVATURES AND STRESSES IN THIN-FILM STRUCTURES ONSUBSTRATES WITH NON-LOCAL EFFECTS” and filed on Mar. 14, 2005, publishedas US 2005-0278126 A1 and PCT/US2005/009406. The U.S. patent applicationSer. No. 11/080,087 further claims the benefits of the U.S. ProvisionalApplications No. 60/576,168 entitled “TECHNIQUES FOR MEASURINGNON-UNIFORM STRESS AND TEMPERATURE IN THIN FILM STRUCTURES” and filed onJun. 1, 2004; and No. 60/614,937 entitled “THE GENERAL CASE OF ARBITRARYCURVATURE AND STRESS VARIATION OF FILM/SUBSTRATE SYSTEMS” and filed onSep. 30, 2004.

The entire disclosures of all of the above-referenced patentapplications and publications are incorporated herein by reference aspart of the specification of this application.

BACKGROUND

This application relates to characterization of structures fabricated onplate substrates including but not limited to integrated structuresfeaturing one or more thin-film layers or graded layers.

Substrates formed of suitable solid-state materials may be used asplatforms to support various structures, such as layered or gradedpanels, and multilevel, thin film microstructures deposited on thesubstrates. Integrated electronic circuits, integrated optical devicesand opto-electronic circuits, micro-electro-mechanical systems (MEMS)deposited on wafers, three-dimensional electronic circuits,system-on-chip structures, lithographic reticles, and flat panel displaysystems (e.g., LCD and plasma displays) are examples of such structuresintegrated on various types of plate substrates. Substrates may be madeof semiconductor materials (e.g., silicon wafers), silicon on insulatorwafers (SOIs), amorphous or glass materials, polymeric or organicmaterials, and others. Different thin material layers or differentstructures may be formed on the same substrate in these structures andare in contact with one another to form various interfaces with adjacentstructures and with the substrate. Some devices may use complexmultilayer or continuously graded geometries. In addition, some devicesmay form various three dimensional structures.

The above and other structures on substrates may be made from amultiplicity of fabrication and processing steps and thus may experiencestresses caused by these steps, such as deposition or thermal stresses.Examples of known phenomena and processes that build up stresses in thinfilms include but are not limited to, lattice mismatch, chemicalreactions, doping by, e.g., diffusion or implantation, rapid depositionby evaporation or sputtering, and material removal (e.g. CMP or etch).As another example, a metallization process in fabricating integratedcircuits may produce multiple layers on a semiconductor substrate (e.g.,silicon), often at elevated temperatures. The multiple layers mayinclude a mixture of metal and dielectric films which usually exhibitdifferent mechanical, physical and thermal properties from those of theunderlying substrate or wafer. Hence, such multiple layers can lead tohigh stresses in the film layers in the interconnection structures.These stresses can cause undesired stress-induced voiding in the metalinterconnects and are directly related to electromigration. In addition,the stresses may cause cracking of some films and even delaminationbetween various film layers, between interconnects and the encapsulatingdielectrics, and between the films and the substrate. It is known thatmetal voiding, electromigration, cracking and delamination are among theleading causes for loss of subsequent process yield and failures inintegrated circuits. Therefore, these and other stresses may adverselyaffect the structural integrity and operations of the structures ordevices, and the lifetimes of such structures or devices. Hence, theidentification of the origins of the stress build-up, the accuratemeasurement and analysis of stresses, and the acquisition of informationon the spatial distribution of such stresses are important in designingand processing the structures or devices and to improving thereliability and manufacturing yield of various layered structures.

Stresses in layered thin-film structures deposited on plate substratesmay be calculated from the substrate curvature or “bow” based on acorrelation between changes in the curvature and stress changes at thesame location. Early attempts to provide such correlation are wellknown. Various formulations have been developed for measurements ofstresses in thin films and most of these formulations are essentiallybased on extensions of Stoney's approximate plate analysis published inProceedings of the Royal Society, London, Series A, vol. 82, pp.172(1909). Stoney used a plate system with a stress bearing, relativelythin film deposited on a relatively thick substrate and derived a simplerelation between the curvature of the plate system and the film stressat the same location based on a linear elasticity for small deformationsand deflections. Stoney's formula also assumed film stresses andcurvatures to be equi-biaxial (i.e., the same in all directions) andspatially constant (i.e., do not change with position) across theplate's surface.

Despite the explicit assumption of spatial uniformity in stress andcurvature, the Stoney formula has often, arbitrarily, been applied toplate systems where this assumption was violated. As an example, theStoney formula was applied in a “pointwise” manner in plate systemswhere the stress and curvature are known to vary with position. Such alocalized application of the Stoney formula was used to extract a“local” value of stress from a “local” value of curvature. Based on this“liberal” interpretation of Stoney's formula, if the curvature componentat any one location of a substrate can be measured, then the film stressat that same location can also be inferred.

SUMMARY

This application describes techniques and associated devices whichinclude non-local effects on stresses in one location contributed byother locations on the same substrate in analysis of non-uniform stressstates in layered or graded film structures on substrates. The presenttechniques and devices were developed in part based on the recognitionthat the uniformity assumption in Stoney's formula with respect to thestresses and curvatures oversimplifies the conditions in many actuallayered or graded structures and devices and, therefore, compromises theaccuracy of the Stoney's formula when applied to such structures. Forexample, according to one implementation described herein, a localcontribution to stresses at a selected location on a layered thin filmstructure formed on a substrate is determined from curvature changes atthe same selected location and in addition from a non-local contributiondetermined from curvature changes at all other locations across thelayered structure. Next, the local contribution and the non-localcontribution are combined to determine the total stresses at theselected location.

In one exemplary method described here, an elastic plate theory analysisis applied to a layered film structure formed on a substrate to includeeffects that the curvatures and stresses of the layered film structureare not spatially uniform and to compute a stress at one location fromcurvature information at the one location and curvature information atother locations. Additionally, a spatial curvature change distributionacross the layered film structure may be obtained and a stress at aselected location from curvature information at the selected locationand curvature information at other locations may be computed accordingto the spatial curvature change distribution.

In another exemplary method described here, a local contribution tostresses at a selected location on a layered film structure formed on asubstrate from curvature changes at the selected location and anon-local contribution to the stresses at the selected location fromcurvature changes at all locations across the layered structure aredetermined. The local contribution and the non-local contribution arecombined to determine the total stresses at the selected location. Aspatial curvature change distribution across the layered structure maybe obtained and used to compute the local contribution and the non-localcontribution.

An example of devices described here includes a module to opticallyinteract with a layered structure and to obtain a full-field curvaturemap of a surface on the layered structure; and a processor incommunication with the module to receive the full-field curvature map.The processor includes means for determining a local contribution tostresses at a selected location on the layered structure from curvaturechanges at the selected location and a non-local contribution to thestresses at the selected location from curvature changes at alllocations across the layered structure, and means for combining thelocal contribution and the non-local contribution to determine the totalstresses at the selected location.

This application also includes a computer-implemented method fordetermining stresses at a location on a layered structure comprising atleast one film formed on a substrate. This method includes applying aspatially varying structural condition in the layered structure tocomputation of stresses at a selected location in the layered structurefrom curvatures at all locations of the layered structure. The spatiallyvarying structural condition in the layered structure includes at leastone of (1) the film covers only a portion of the substrate, (2) the filmhas a film thickness that varies from one location to another across thefilm, and (3) the substrate has a substrate thickness that varies fromone location to another. This method further includes determining in thecomputation under the spatially varying structural condition a localcontribution to stresses at the selected location on the layeredstructure from curvature changes at the selected location; determining anon-local contribution to the stresses at the selected location fromcurvature changes at all locations of the layered structure; andcombining the local contribution and the non-local contribution toestimate the total stresses at the selected location.

This application also describes a method for monitoring a substratefabrication process using the above computer-implemented method. Whilethe layered structure is being processed, an optical probe beam isdirected to the layered structure to optically obtain a full-fieldcurvature map of the layered structure. The full-field curvature map isprocessed to obtain curvature information at all locations of thelayered structure. This method further includes applying thecomputer-implemented method to determine the total stresses at eachlocation of the layered structure; and determining whether the layeredstructure is defective based on an acceptable threshold stress.

These and other implementations, and associated advantages are nowdescribed in greater detail in the following figures, the detaileddescription, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows an example of a device for measuring full-field curvaturemaps of a sample and processing the map to analyze the stresses on thesample through a non-local analysis.

FIG. 1B illustrates various film stress components.

FIG. 1C show CGS full field measurements of curvatures of thex-direction, y-direction and mixed curvature distributions of a large,industrial wafer composed of a low-k dielectric film deposited on a Sisubstrate.

FIG. 1D is a schematic diagram of the thin film/substrate system,showing the cylindrical coordinates (r, θ, z).

FIG. 1E shows full field curvature measurements of maximum and minimumprincipal curvatures of a thin film/substrate system using a CGS system.

FIG. 1F is a schematic diagram of the non-uniform shear tractiondistribution at the interface between the film and the substrate.

FIG. 1G shows a schematic diagram of a thin film/substrate system inn anisland configuration where a film layer and the underlying substratehave radii.

FIG. 1 shows a system with a shearing device for measuring a surfaceaccording to one implementation.

FIG. 2 shows an interference pattern to illustrate a phase shiftingtechnique.

FIGS. 3 and 4 show two coherent gradient sensing (CGS) systems formeasuring surfaces.

FIGS. 5A and 5B show two exemplary phase shifting techniques in CGS.

FIGS. 6A, 6B, 7A, 7B, 7C, 8, 9, 10A, 10B, 11A, 11B, and 11C showexamples of non-CGS shearing interferometers suitable for measuringsurfaces based on the techniques described in this patent application.

FIG. 12 illustrates an exemplary layout for optically measuring thebackside surface of a wafer where wafer supports are in contact with thebackside surface.

FIG. 13 illustrates an example where the backside of the wafer issupported by three wafer supports that are oriented in a non-symmetricway to enable direct collection of data on the full wafer surface bymaking multiple measurements of the wafer at different angularorientations.

DETAILED DESCRIPTION

Stoney's formula was derived for an isotropic “thin” solid film ofuniform thickness deposited on a much “thicker” plate substrate based ona number of assumptions. Stoney's assumptions include the following: (1)deformations and deflections in the plate system are small; (2) therelation between stresses and strains is linear as in Hook's law; (3)the film stress states are in-plane isotropic or equi-biaxial (two equalstress components in any two, mutually orthogonal in-plane directions)while the out of plane direct stress and all shear stresses vanish; (4)the system's curvature change components are equi-biaxial (two equaldirect curvatures) while the twist curvature vanishes; and (5) allsurviving stress and curvature components are spatially constant overthe plate system's surface, a situation which is often violated inpractice.

The assumption of equi-biaxial and spatially constant curvature isequivalent to assuming that the plate system would deform sphericallyunder the action of the film stress. If this assumption were to be true,a rigorous application of Stoney's formula would indeed furnish a singlefilm stress value. This value represents the common magnitude of each ofthe two direct stresses in any two, mutually orthogonal directions. Thisis the uniform stress for the entire film and it is derived frommeasurement of a single uniform curvature value which fullycharacterizes the system provided the deformation is indeed spherical.

Despite the explicitly stated assumptions of spatial stress andcurvature uniformity, the Stoney formula is often, arbitrarily, appliedto cases of practical interest where these assumptions are violated.This is typically done by applying Stoney's formula pointwise and thusextracting a local value of stress from a local measurement of thecurvature of the system. This approach of inferring film stress clearlyviolates the uniformity assumptions of the analysis and, as such, itsaccuracy as an approximation is expected to deteriorate as the levels ofcurvature non-uniformity become more severe. To the best knowledge ofthe inventors, no analytical formulation capable of dealing withnon-uniform stress and deformation states has been in existence.

Following the initial formulation by Stoney, a number of extensions havebeen derived by various researchers who have relaxed some of the otherassumptions (other than the assumption of uniformity) made by hisanalysis.

Such extensions of the initial formulation include relaxation of theassumption of equi-biaxiality as well as the assumption of smalldeformations/deflections. A biaxial form of Stoney, appropriate foranisotropic film stresses, including different stress values at twodifferent directions and non-zero, in-plane shear stresses, was derivedby relaxing the assumption of curvature equi-biaxiality. Relatedanalyses treating discontinuous films in the form of bare periodic linesor composite films with periodic line structures (e.g. encapsulatedlines) have also been derived. See, U.S. Pat. No. 6,600,565 entitled“REAL-TIME EVALUATION OF STRESS FIELDS AND PROPERTIES IN LINE FEATURESFORMED ON SUBSTRATES” and issued to U.S. Pat. No. 6,513,389 “TECHNIQUEFOR DETERMINING CURVATURES OF EMBEDDED LINE FEATURES ON SUBSTRATES” toSuresh and Park. These latter analyses have also removed the assumptionof equi-biaxiality and have allowed the existence of three independentcurvature and stress components in the form of two, non-equal, directand one shear or twist component. However, the uniformity assumption ofthese quantities over the entire plate system was retained.

In addition to the above, single, multiple and graded films andsubstrates have been treated in a “large” deformation analysis. See,U.S. Pat. No. 6,781,702 entitled “DETERMINING LARGE DEFORMATIONS ANDSTRESSES OF LAYERED AND GRADED STRUCTURES TO INCLUDE EFFECTS OF BODYFORCES” and issued to Giannakopoulos et al. The analysis techniquedescribed in this patent removes the restriction of an equi-biaxialcurvature state and the assumption of “small” deformations ordeflections. This analysis allows for the prediction of kinematicallynon-linear behavior and bifurcations in curvature states. Thesebifurcations are transformations from an initially equi-biaxial to asubsequently biaxial curvature state that may be induced by an increasein film stress beyond a critical level. This critical level isintimately related to the systems aspect ratio, i.e., the ratio ofin-plane to thickness dimensions and elastic stiffness. This analysisalso retains the assumption of spatial curvature and stress uniformityacross the system. However, it allows for deformations to evolve from aninitially spherical shape to an energetically favored state whichfeatures three different, still spatially constant, curvature components(e.g. ellipsoidal, cylinder or saddle shape).

None of the above-discussed extensions of Stoney's methodology hasrelaxed the most restrictive of Stoney's original assumption ofuniformity which does not allow film stress and curvature components tovary across the plate surface.

This crucial assumption is often violated in practice since filmstresses and the associated system curvatures are non-uniformlydistributed over the plate area. This is often true due tonon-uniformities in thermal processing (e.g. non-uniform temperaturedistributions during heating or cooling) and due to othernon-uniformities resulting during the film deposition processes.

The present techniques remove this very restrictive assumption and allowfor accurate inference of spatially varying film stress components fromfull field measurement of non-uniform curvature components performedacross the entire plate system. Unlike Stoney and the above-describedextensions, the present analysis shows that the dependence of stress oncurvature is non-local. Hence, the stress at a point on the film maydepend on both the local value of curvature (at the same point) and onthe value of curvatures of all other points on the plate system(non-local dependence). The more pronounced the curvaturenon-uniformities are, the more important non-local effects become inaccurately determining film stresses.

This demonstrates that other techniques based on Stoney's analysis andvarious extensions which do not allow for non-uniformities cannot handlethe non-locality of the stress/curvature dependence. This will result insubstantial prediction errors if such analyses are applied locally incases where spatial variations of system curvatures are observed.

The techniques and devices described here use spatially-varyingcurvature information, gathered in a full field and real time manner, todetermine stresses at each location from both local and non-localcontributions from other locations. The specific form of both local andnon-local contributions to be described later is not ad hoc. It has beenrigorously derived by means of an elastic plate theory formulation whichincludes among other things strict enforcement of the appropriate stressequilibrium conditions at both the thin film and the substrate and ofthe appropriate continuity (tractions and displacements) condition ofthe film substrate interface as well as enforcement of plate boundaryconditions at the edge of the plate for both the film and the substratesupporting the film. Notably, the existence of important non-localcontributions, predicted by this analysis, necessitates the use of afull field technique for the measurement of all curvature componentsover the entire surface of the plate system. This is because the stressstate at a point depends on curvature contributions from the entiresystem.

FIG. 1A shows an example of a device for measuring stresses in a layeredplate structure with the above non-local contribution. The layeredstructure includes one or more thin films formed on a thick substrateand may be held in a sample holder. An instrument or module formeasuring all three of the full-field curvature component maps of onesurface of the layered plate structure is implemented to probe thelayered plate structure to perform the curvature measurement. As anexample, the instrument may optically interact with the layeredstructure to measure the surface via one or more optical beams. Asdescribed in this application, the instrument may include an opticalinterferometer such as a shearing interferometer. The device in FIG. 1Afurther includes a processor that receives and processes the threefull-field curvature component maps with two direct curvatures and thetwist curvature and produces the stresses on the layered structure undermeasurement with both local and non-local contributions. The processoris programmed with suitable processing software instructions or routinesthat are described in this application. The device in FIG. 1A may beused for in-situ and real-time monitoring of a substrate or wafer underfabrication. As illustrated, the wafer may be located in a processingchamber while being measured by, e.g., an optical probe beam. Details ofvarious exemplary implementations of the device in FIG. 1A are describedbelow.

The following sections first describe techniques for analyzing and usingrelations between film stresses and system curvatures in layered platestructures that have radially symmetric distribution of film stresses orwafer temperature. Next, the techniques for analyzing relations betweenfilm stresses and system curvatures in layered plate structures witharbitrary spatial variations of film stresses, curvatures, and wafertemperature are presented. In addition, relations between film stressesand temperature as well as curvature and the temperature are described.

Radially symmetric or axisymmetric variations are often present invarious layered structures. This is in part due to the inherentnon-uniformities and imperfections in the fabrication processes used formaking such structures. For example, typical semiconductor wafers andlayered structures formed on such wafers frequently exhibit radiallysymmetric variations in curvatures and stresses. Such variations may becaused by radial temperature variations associated with the thermaltreatment of circular wafers both during the cooling or heating phasesof a thermal processing step. Such variations may also be caused bynon-uniformities in the film deposition process which may result in theestablishment of radial stress gradients across the film surface.

Let K₁₁, K₂₂, K₁₂ be the three independent Cartesian components ofcurvature (two direct and one twist) in a coordinate system x₁, x₂. Thiscoordinate system is used as a reference and defines the plane of acircular plate structure in an un-deformed state prior to thedevelopment of the curvature. The principal maximum and minimumcurvatures in terms of K₁ and K₂ are given by: $\begin{matrix}{K_{1,2} = {\frac{K_{11} + K_{22}}{2} \pm \left\{ {\left( \frac{K_{11} - K_{22}}{2} \right)^{2} + K_{12}^{2}} \right\}^{1/2}}} & (1)\end{matrix}$It can be shown that for radially symmetric or axisymmetric deformationsthe principal directions of curvature are radial and circumferential (orazimuth) lines and that the principal curvature fields are equal toK_(rr)(r) (along the radial coordinate direction) and K_(θθ)(r) (alongthe orthogonal azimuth coordinate direction), respectively. Theseprincipal curvatures are functions of radial distance, r, from the wafercenter. The polar twist curvature tensor component K_(rθ)(r) vanishes atevery radial location on the circular plate.

The two curvature invariants of interest here are: $\begin{matrix}{{I = {\frac{K_{1} + K_{2}}{2} = \frac{K_{rr} + K_{\theta\theta}}{2}}}\quad{and}{J = \frac{K_{rr} - K_{\theta\theta}}{2}}} & (2)\end{matrix}$where${{K_{rr}(r)} = \frac{\partial^{2}f}{\partial r^{2}}},\quad{{K_{\theta\theta}(r)} = {\frac{1}{r}\frac{\partial f}{\partial r}}}$and f(r) is the equation describing the radial topography of the platestructure under consideration. For the most general case of radialsymmetry, the two curvature components and the two invariants arefunctions of the radial position on the wafer. It should be noted thatfor idealized system shapes of the spherical type such as the onesassumed in the derivation of the Stoney's formula, the followingrelations exist:K₁₁=K₂₂=K_(rr)=K_(θθ)=K, and K₁₂=K₂₁=0, K_(rθ)=0   (3)For these special cases, a single curvature number, K, which is alsoconstant across the wafer, is sufficient to define the entire shape.Accordingly, the invariants I and J defined reduce to:I=K, J=O   (4)This is a very restrictive and special case of radial symmetry which isseldom observed in practical structures used in devices.

In many practical situations, various industrial processing steps, suchas wafer anneal, wafer cooling by, e.g., either in a rapid thermalprocessing (RTP) or batch furnaces, involve radial temperature fields(albeit time varying) because of chamber and wafer geometry which oftenimposes such a radially symmetric condition. The resulting thermalstresses associated with mismatches of the coefficients of thermalexpansion (CTE) between film and wafer substrate are also radiallysymmetric. The same is often true within film deposition chambers wherefilm deposition by radially varying gas flow may impose radial stress(intrinsic) symmetry. In these and other situations, the presence ofradially symmetric stress fields, either thermally induced or intrinsic,can lead to wafer curvatures that are also radially symmetric.

Consider a circular composite plate structure which includes a thermallyand mechanically isotropic thin film with a uniform thickness formed onan isotropic substrate with a uniform but much larger thickness. Thefollowing notations are used for convenience. Notations of h_(f) andh_(s) represent film and substrate thickness, respectively. Thenotations of α_(f), α_(s), E_(f), E_(s), ν_(f), ν_(s) are used torepresent the coefficient of thermal expansion (CTE), Young's modulus,Poisson's Ratio of film and substrate, respectively, where the subscript“s” denotes the substrate and the subscript “f” denotes the thin film onthe substrate. The composite plate structure is subject to a non-uniformbut radially symmetric increase in temperature T(r) measured over auniform reference state. The temperature field and the resultingcurvature and stress fields may vary with the radial position, r, andwith time: T=T(r, t). The dependence of these quantities on the time (t)is implied in the following description but not explicitly shown. It isassumed that the substrate is circular with a radius of R. Computationsfor rectangular and other geometries can be readily derived based on thetechniques described herein. Other related notations are as listedbelow:

E _(f,s)=E_(f,s)/(1−v_(f,s)) are the film and the substrate biaxialmoduli, respectively; Ê_(f,s)=E_(f,s)/(1−v² _(f,s)) are the film and thesubstrate plane strain moduli, respectively.

T(r) is the radial temperature increase profile and may vary with time;${\overset{\_}{T}(r)} = {\frac{2}{r^{2}}{\int_{o}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}}$is the averaged temperature over a disc area which is centered at thewafer center and has a radius r<R.${\overset{\_}{T}(R)} = {\frac{2}{R^{2}}{\int_{o}^{R}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}}$is the instantaneous averaged temperature over the entire wafer.

T(r)− T(r) is the deviation of the local temperature from the averagedtemperature over the disk from the center to a radius of r; and T(r)−T(R) is the deviation of the instantaneous local temperature from thewafer averaged temperature.

The radially varying temperature increase T(r) imposed on the compositeplate structure establishes thermal mismatch stresses on the film. Thesethermal mismatch stresses in turn deform the composite plate structureand establish a radially symmetric deformation with two distinct,radially symmetric principal curvatures K_(rr)(r) and K_(θθ)(r).

Based on linear thermoelasticity and the plate theory for thin filmstructures, the following relations between K_(rr)(r), K_(θθ)(r) andT(r) can be derived: $\begin{matrix}{{K_{\theta\theta}(r)} = {\frac{6{\hat{E}}_{f}h_{f}}{{\hat{E}}_{s}h_{s}^{2}}\begin{Bmatrix}{{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack\frac{1}{r^{2}}{\int_{0}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}} +} \\{\begin{bmatrix}{{\frac{1 - v_{s}}{\quad{1 + v_{\quad s}}}\left( {1 + v_{f}} \right)\left( {\alpha_{s} - \alpha_{f}} \right)} -} \\{\alpha_{s}\left( {v_{s} - v_{f}} \right)}\end{bmatrix}\frac{1}{R^{2}}{\int_{0}^{R}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}}\end{Bmatrix}}} & (5) \\{{K_{rr}(r)} = {\frac{6{\hat{E}}_{f}h_{f}}{{\hat{E}}_{s}h_{s}^{2}}\begin{Bmatrix}{{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack{T(r)}} -} \\{{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack\frac{1}{r^{2}}{\int_{0}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}} +} \\{\begin{bmatrix}{{\frac{1 - v_{s}}{\quad{1 + v_{\quad s}}}\left( {1 + v_{f}} \right)\left( {\alpha_{s} - \alpha_{f}} \right)} -} \\{\alpha_{s}\left( {v_{s} - v_{f}} \right)}\end{bmatrix}\frac{1}{R^{2}}{\int_{0}^{R}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}}\end{Bmatrix}}} & (6)\end{matrix}$It should be noted that, unlike the Stoney-like formulations, the aboveexpressions involve integrals over r and are non-local in nature. Hence,at a radical location r, the curvatures K_(rr)(r) and K_(θ74) (r) aredependent not only on the local value of the temperature T(r) but alsoon the integrals of T(r) which are evaluated from zero to r or from zeroto R. These radial integrals represent the contributions to curvature ata fixed location resulting from the temperature increases at neighboringlocations (non-locality). Also, it is noted that these expressionscannot be directly inverted to obtain temperature from the curvaturecomponents. Similar expressions can be derived for the Cartesiancomponents K₁₁(r,θ), K₂₂(r,θ) and K₁₂(r,θ) which are functions of θ.

Based on the above, the wafer temperature and curvature invariants cannow be written as follows: $\begin{matrix}\begin{matrix}{I = \frac{K_{rr} + K_{\theta\theta}}{2}} \\{= {\frac{6{\overset{\_}{E}}_{f}h_{f}}{{\overset{\_}{E}}_{s}h_{s}^{2}} \cdot \begin{Bmatrix}{{\left( {\alpha_{\quad s} - \alpha_{\quad f}} \right){T(r)}} +} \\{\left\lbrack {{\alpha_{\quad s}\left\lbrack {\frac{\quad\left( {1\quad + \quad v_{\quad s}} \right)^{2}}{\quad{2\quad\left( {1\quad + \quad v_{\quad f}} \right)}} - 1} \right\rbrack} + {\frac{\alpha_{f}}{2}\left( {1 - v_{s}} \right)}} \right\rbrack\left( {{T(r)} - {\overset{\_}{T}(R)}} \right)}\end{Bmatrix}}}\end{matrix} & (7)\end{matrix}$

The above invariant I has the following properties:

-   -   The first term is Stoney-like with a temperature increase        replaced by the local value of T(r) at any radial distance        (local contribution).    -   The second term is proportional to [T(r)− T(R)] which is the        deviation of the local temperature from the averaged temperature        over the entire wafer (non-local contribution).    -   When T(r)=T_(c)=constant, only the first term exists and        Stoney's approximate expressions are derived as a special case.        In addition, under T(r)=T_(c), Equations (5) and (6) give the        following relation:        $K_{rr} = {K_{\theta\theta} = {\left\lbrack \frac{6{\hat{E}}_{f}h_{f}}{h_{s}^{2}} \right\rbrack\left( {\alpha_{s} - \alpha_{f}} \right)T_{c}}}$        which is the equi-biaxial Stoney result.    -   The invariant I is the local mean curvature at a point r.

Taking radial averages over both sides of Eq. (7), the averagetemperature T(R) over the entire wafer surface can be expressed as afunction of the wafer averaged means curvature I(R): $\begin{matrix}{{{{\overset{\_}{T}(R)} = {\frac{{\overset{\_}{E}}_{s}h_{s}^{2}}{6{\overset{\_}{E}}_{f}{h_{f}\left( {\alpha_{s} - \alpha_{f}} \right)}} \cdot {\overset{\_}{I}(R)}}},{where}}{{\overset{\_}{I}(R)} = {\frac{1}{R^{2}}{\int_{0}^{R}{\eta\quad{I(\eta)}{\mathbb{d}\eta}}}}}} & (8)\end{matrix}$

-   -   It should be noted that measurement of the wafer averaged mean        curvature I(R) provides the average temperature T(R).    -   Substitution of Eq. (8) into Eq.(7) provides a relation between        the local value of the temperature T(r) and the local value of        the mean curvature I(r) as follows: $\begin{matrix}        {{T(r)} = {\frac{{\overset{\_}{E}}_{s}h_{s}^{2}}{6{\overset{\_}{E}}_{f}h_{f}}\frac{1 + v_{f}}{1 + v_{s}}\begin{Bmatrix}        \frac{2}{\quad{{\alpha_{s}\left( {1 + v_{s}} \right)} - {\alpha_{f}\left( {1 + v_{f}} \right)}}} \\        {\left\lbrack {\frac{\quad{{K_{rr}(r)} + {K_{\theta\theta}(r)}}}{2} - {\overset{\_}{I}(R)}} \right\rbrack +} \\        {\frac{1 + v_{s}}{1 + v_{f}}\frac{1}{\alpha_{s} - \alpha_{f}}{\overset{\_}{I}\left( R \right.}}        \end{Bmatrix}}} & (9)        \end{matrix}$

The above relation in Eq. (9) enables the full field inference of thetemperature profile, T(r), provided that the mean curvatureI(r)=(K_(rr)+K_(θθ))/2 can be measured and that the film and substrateproperties α_(s), α_(f), E_(s), E_(f) and ν_(s), ν_(f) are known. Theavailability of full field methods for measuring I(r) is preferable butis not required for enabling this methodology. For time varying and,radially symmetric temperature profiles T(r,t), the methodology remainsidentical. The existence of a curvature measurement method that can beimplemented in real-time is needed.

Equations (5) and (6) can now be used to evaluate the second curvatureinvariant J as follows: $\begin{matrix}{J = {\frac{K_{rr} - K_{\theta\theta}}{2}\quad = {{\frac{3\quad{\hat{E}}_{f}h_{f}}{{\hat{E}}_{s}h_{s}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack}\left\{ {{T(r)} - {\overset{\_}{T}(r)}} \right\}}}} & (10)\end{matrix}$It is noted that, when T(r)=T_(c)=Constant, J vanishes, and that theabove equation is non-local, with integrals over r, and thus is notStoney-like. When the temperature is uniform, the expression for Jyields K_(rr)=K_(θθ).

The solution of the radially symmetric thermoelastic film/substrateplate structure also furnishes relations between the direct, in-plane,film stress components and T(r): $\begin{matrix}{{\frac{\sigma_{rr} + \sigma_{\theta\theta}}{2} = {\frac{E_{f}}{1 - v_{f}}\left\{ {{\left( {\alpha_{s} - \alpha_{f}} \right){T(r)}} - {\frac{\left( {1 - v_{s}} \right)}{2}{\alpha_{s}\left( {{T(r)} - {\overset{\_}{T}(R)}} \right)}}} \right\}}},} & (11) \\{\frac{\sigma_{rr} - \sigma_{\theta\theta}}{2} = {\frac{1}{2}\frac{E_{f}}{\left( {1 + v_{f}} \right)}\left( {1 + v_{s}} \right)\alpha_{s}\left\{ {{T(r)} - {\overset{\_}{T}(r)}} \right\}}} & (12)\end{matrix}$The following properties of equations (11) and (12) are noted here.

-   -   Both equations (11) and (12) feature a non-local part.    -   The first term on the right hand side of equation (11) is a        local, Stoney-like, term and involves the local dependence of        stress on T(r). The second term is non-local and depends on the        deviation of T(r) from its average value T(R). This term        vanishes when T(r)=T_(c)=constant.    -   Equation (12) is entirely non-local. For a spatially uniform        temperature increase distribution T(r)=T_(c)=const, Eq. (12)        predicts an equi-biaxial and spatially uniform state        σ_(rr)=σ_(θθ), consistent with Stoney's restrictive assumptions.    -   In general, Eqs. (11) and (12) for the sum and the difference of        the film stresses can be used to solve for the individual        components σ_(rr) and σ_(θθ). In the general case of radial        non-uniformity, these stresses will not be equal to each other        at each point and their ration will vary radially from point to        point.

In addition to the direct, in-plane film stress components, the radiallysymmetric plate analysis also furnishes expressions for the shear stressacting at the interface between the film and the substrate, along theradial direction. This component is denoted by τ_(r) and followingconventional notations are given by τ_(r)=σ_(3r)=σ_(r3) where thesubscript, 3, denotes the x₃ direction perpendicular to thefilm/substrate interface. For the radially symmetric case the shearstress acting on this interface along the circumferential (or azimuth)direction vanishes because of the radial symmetry(τ_(θ)=σ_(3θ)=σ_(θ3)=0). The interfacial shear stress along the radialdirection is given by: $\begin{matrix}{\tau_{r} = {\sigma_{3r} = {\sigma_{r\quad 3} = {\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\left\{ {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\}\frac{\mathbb{d}T}{\mathbb{d}r}}}}} & (13)\end{matrix}$

It should be noted that the right hand side of equation (13) isproportional to the radial derivative of the temperature. As a result,the interfacial shear stress is shown to arise because of specialnon-uniformities. When T(r)=T_(c)=constant, the shear stress τ_(r)vanishes and the analysis reduces, as a special case, to the restrictivecase treated by Stoney in which no shear stresses appear. In addition,since interfacial shear stresses are known to cause delamination betweenthin films and substrates, the relation in Eq. (13) is of particularpractical importance in the failure analysis of such systems.

In the previous sections, the relations between system curvatures andtemperature as well as film stresses and temperature have beenestablished. Elimination of temperature from such relations can providerelations between the film stress and the system curvature components.

Eqs. (5), (6), (8), (11) and (12) may be used to eliminate thetemperature terms to provide the following stress-curvature invariantrelations: $\begin{matrix}{\frac{\sigma_{rr} + \sigma_{\theta\theta}}{2} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\begin{Bmatrix}{\frac{K_{rr} + K_{\theta\theta}}{2} +} \\{\begin{bmatrix}{\frac{1 - v_{s}}{1 + v_{s}} -} \\\frac{\alpha_{s}\left( {1 - v_{f}} \right)}{{\alpha_{s}\left( {1 + v_{s}} \right)} - {\alpha_{f}\left( {1 + v_{s}} \right)}}\end{bmatrix} \times} \\\left\lbrack {\frac{K_{rr} + K_{\theta\theta}}{2} - {\overset{\_}{I}(R)}} \right\rbrack\end{Bmatrix}}} & (14) \\{\frac{\sigma_{rr} - \sigma_{\theta\theta}}{2} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\frac{\alpha_{s}\left( {1 - v_{f}} \right)}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}}\frac{\left( {K_{rr} - K_{\theta\theta}} \right)}{2}}} & (15)\end{matrix}$Based on the above, individual stress components can be evaluated byadding and subtracting equations (14) and (15) and are given in terms ofI(r), J(r) and I(R).

Equation (14) includes two terms. The first term is local andStoney-like, and depends on the local value of the curvature invariant${I(r)} = \frac{{K_{rr}(r)} + {K_{\theta\theta}(r)}}{2}$which can be estimated when the sum of (K_(rr)+K_(θθ)) can be measuredat each location. The second term depends on the local differencebetween the local value of I(r) and its wafer averaged value${\overset{\_}{I}(R)} \equiv {\frac{2}{R^{2}}{\int_{0}^{R}{\left\lbrack \frac{{K_{rr}(\eta)} + {K_{\theta\theta}(\eta)}}{2} \right\rbrack\eta{\mathbb{d}\eta}}}}$Both I(r) and I(R) can directly be measured provided that a full fieldmeasurement technique, such as an optical full-field measurement, isavailable.

Equation (15) relates to (σ_(rr)−σ_(θθ))/2 to${J(r)} = {\frac{{K_{rr}(r)} = {K_{\theta\theta}(r)}}{2}.}$Unlike the relation in Eq. (14), the relation in Eq. (15) is purelylocal and is non-trivial and has a non-zero right hand side only forradially varying stresses. For spherical deformations in Stoney's model,Eq. (15) predicts equi-biaxial stress states (σ_(rr)=σ_(θθ)). Foraxially symmetric cases, σ_(rr) and σ_(θθ) are principal stresses andtheir difference divided by two is equal to the maximum in-plane shearstress. Equating this maximum shear stress to a yield stress τ_(y) isequivalent to implementing the Hüber-Von Mises yield criterion. Eq. (14)can now be used to establish a critical level of J(r)) for which plasticyield would commence in the film (i.e., τ_(max)=(σ_(rr)−σ_(θθ))/2=τ_(y)(yield stress)). If such a level is exceeded, the locations on the filmfor which plasticity becomes a problem can be identified.

To successfully implement equations (14) and (15), both K_(rr) andK_(θθ) should be measured in full field and, for time-varyingcurvatures, in real time. For radial symmetry these curvatures areprincipal curvatures and can be calculated from Eq. (1) following themeasurement of K₁₁, K₂₂, and K₁₂.

Equation (2) provides the invariants I(r) and J(r) that are needed inthe equation. In addition to I(r) and J(r) the wafer averaged meancurvature${\overset{\_}{I}(R)} \equiv {\frac{2}{R^{2}}{\int_{0}^{R}{{I(\eta)}\eta{\mathbb{d}\eta}}}}$needs to be computed from the full field curvature data such ascurvature data optically obtained with one of the optical techniquesdescribed in this application.

The above section has examined the class of radially symmetricdistributions of film membrane force f_(r)(r) and f_(θ)(r) associatedwith arbitrary radial variations of film stress, σ_(rr)(r) andσ_(θθ)(r), resulting from a non-uniform wafer temperature increase,T(r), or from other forms of non-uniform mismatch strain associated withimperfect deposition processes. The radial temperature variation hasbeen used as a specific example to illustrate the techniques and toevaluate K_(rr)(r) and K_(θθ)(r) in terms of T(r), T(r) and T(R) inEquations (5) and (6). These relations are found to be non local andnon-Stoney like.

The system curvature invariant (K₁₁+K₂₂)/2=(K_(rr)+K_(θθ))/2 (the meancurvature) can be expressed as the sum of two terms (eq. (7)). The firstterm is identical to the Stoney prediction with the constant temperatureincrease replaced by the local value of temperature at any radialdistance. The second term, which can be viewed as a correction to or adeviation from the Stoney prediction, is proportional to the deviationof the local temperature from the average temperature (averaged over theentire wafer). The first term is local while the second one isnon-local.

Since the film stresses can also be expressed as two terms (one localand one non local) proportional to the local temperature and the localdeviation from the average of the mean respectively (Eq. (11) and (12)),the elimination of the temperature can provide a relation between a filmstress invariant (radially varying mean stress) and a system curvatureinvariant (radially varying mean curvature) as well as its averagedvalue over the wafer area. This is given in Eq. (14).

A similar relation between another curvature invariant (maximum twist)and another film stress invariant (max shear stress) can also beobtained. The relation in Eq.(15) is purely local unlike the oneinvolving mean stress and curvature.

The individual stress components σ_(rr)/σ_(θθ) can now be expressed interms of the curvature invariants and the wafer area averaged meancurvature. These formulas are the non-local generalizations of theStoney's formula appropriate for non-uniform radially symmetricdeformations and can be used to analyze the entire class of radialvariations of random shape and provide the means of using full fieldmeasurements to calculate spatially varying biaxial stresses pointwisethrough various types of metrology capable of providing full field, realtime measurement of curvature components and their invariants, such asoptical metrology (e.g. any type of shearing or non-shearinginterferometry such as CGS or Twyman-Green interferometers).

Evaluation of the maximum twist allows for the measurement of themaximum, in-plane shear stress and the establishment of the radialregions on the wafer where a thin film may reach plastic flow. This isdone by requiring that the maximum shear stress be equal to the filmyield stress as required by the Huber-Von Mises plastic flow criterion.

Inverse implementation of the H.R. formula and the “yield criterion” mayallow for the calculation of critical levels of temperature variationsfrom the mean, or of critical levels of twist necessary to produceplastic yielding of the film. Since plastic flow is generallyundesirable this will provide a means of setting anneal temperaturevariation thresholds (e.g. in either RTP or batch furnaces) to avoidyielding.

The curvature/temperature versions of these formulas can be used tomeasure the radial profile of a heated or cooled wafer (in either anin-situ or ex-situ environment) in real time through appropriateinterferometric measurement of curvature invariants in Eqs. (8) and (9).

Elimination of the temperature from equations (13) and (9) provides arelation between the interfacial shear stress τ_(r)=σ_(3r)=σ_(r3)(acting along the radial direction) and the radial gradient of the firstcurvature invariant as follows: $\begin{matrix}{\tau_{r} = {\sigma_{3r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{x}^{2}} \right)}\frac{\mathbb{d}}{\mathbb{d}r}\left( {K_{rr} + K_{\theta\theta}} \right)}}} & (16)\end{matrix}$The above equation is completely non-local in nature.

Since interfacial shear stresses are responsible for promoting systemfailures through delamination of the thin film from the substrate, Eq.(16) has particular significance. It shows that such stresses areproportional to the radial gradient of K_(rr)+K_(θθ) and not to it'smagnitude as might have been expected of a local, Stoney-likeformulation. The implementation value of Eq. (16) is that it provides aneasy way of inferring these special interfacial shear stresses once thefull field curvature information is available. As a result, themethodology also provides a way to evaluate the risk of and to mitigatesuch important forms of failure. It should be noted that for the specialcase of constant curvatures, this shear stress, τ_(r), vanishes as isthe case for all Stoney-like formulations described in the introduction.

The above curvature-invariant/stress-invariant relations in radiallysymmetric layered structures can be generalized to non radiallysymmetric deformations. The techniques for analyzing layered platestructures beyond the class of geometries with a radial symmetry aredescribed below and are applicable to arbitrary variations of stressesand curvatures across the system (e.g. wafer or other film substratesystem, or other layered plate structure).

The problem under consideration is still a thermally and mechanicallyisotropic thin film (or thin film structure) which is deposited on amuch thicker circular substrate. In this very general case the film cansustain arbitrary spatial variations of stresses which can vary withboth radial and angular positions. This non-uniform stress state hasbeen established either by a non-uniform temperature distribution actingover the entire plate system or by other reasons relating to thedeposition process or other processing steps. The situation with anon-uniform temperature over the entire plate system is considered hereas an example.

Additional notations are introduced here for use in the followingsections for describing the techniques for analyzing the general case.More specifically, T(r,θ,t) represents an arbitrary temperature increaseprofile (measured over a uniform ambient), which may be a function ofthe radial and angular positions (r, θ), and may also be a function oftime. The film-substrate system is still assumed to be a circular diskwith a radius of R. The spatially averaged temperature over the entiredisk substrate with an area of A and a radius R is:${\overset{\_}{T}\left( {R,t} \right)} = {{\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{{T\left( {r,\theta,t} \right)}{\mathbb{d}A}}}}} = {\frac{1}{\pi\quad R^{2}}{\int_{0}^{R}{\int_{0}^{2\pi}{{T\left( {\eta,\theta,t} \right)}\eta{\mathbb{d}\eta}{\mathbb{d}\theta}}}}}}$which is reduced to the definition of T(R) for the axisymmetric geometrydescribed previously. Accordingly, the difference of T(r,θ,t)− T(R)represents the deviation of the local temperature field, which varieswith r and θ, from the wafer averaged 5 temperature. Notations for thesystem's thickness, thermal and mechanical properties h_(f), h_(s),α_(f), α_(s), E_(f), E_(s), ν_(s)/ν_(f), E _(f,s), and Ê_(f,s) aredefined as before. All fields are expressed in Polar coordinates r and θdenoting position on the film substrate system. The arbitrarytemperature profile T(r,θ,t) is expressed in polar coordinates and,without loss of generality, can be expanded in Taylor series as follows:$\begin{matrix}{{T\left( {r,\theta,t} \right)} = {{T^{(0)}\left( {r,t} \right)} + {\sum\limits_{n = 1}^{\infty}{{T_{c}^{(n)}\left( {r,t} \right)}\cos\quad n\quad\theta}} + {\sum\limits_{n = 1}^{\infty}{{T_{s}^{(n)}\left( {r,t} \right)}\sin\quad n\quad\theta}}}} & (17)\end{matrix}$The coefficients of the Taylor expansion are then given by:$\begin{matrix}{{{T^{(0)}\left( {r,t} \right)} = {\frac{1}{2\pi}{\int_{0}^{2\pi}{{T\left( {r,\theta,t} \right)}{\mathbb{d}\theta}}}}}{{{T_{c}^{(n)}\left( {r,t} \right)} = {\frac{1}{\pi}{\int_{0}^{2\pi}{{T\left( {r,\theta,t} \right)}\cos\quad n\quad\theta{\mathbb{d}\theta}}}}},{n = 1},2,3}{{T_{s}^{(n)}\left( {r,t} \right)} = {\frac{1}{\pi}{\int_{0}^{2\pi}{{T\left( {r,\theta,t} \right)}\sin\quad n\quad\theta{\mathbb{d}\theta}}}}}} & (18)\end{matrix}$Any arbitrary temperature variation can be expressed in terms of aninfinite series as described above.

The system curvature (e.g. K_(rr), K_(θθ), K_(rθ)) components, thestress (e_(rr), σ_(θθ), σ_(rθ), σ_(3r)/ σ_(3θ) . . . ) components andthe temperature increase field T. all are arbitrary functions ofin-plane position (r,θ) and time (t). We first provide the generalizedrelations between the system curvatures and T. In the followingrelations, the dependence on r, θ and t is implied but not explicitlyshown. The variations are completely general. The expression for the sumof the direct curvature is given by $\begin{matrix}{{K_{rr} + K_{\theta\theta}} = {\frac{6E_{f}{h_{f}\left( {1 - v_{s}^{2}} \right)}}{E_{s}{h_{s}^{2}\left( {1 - v_{f}^{2}} \right)}}\left\langle {{2\left( {\alpha_{s} - \alpha_{f}} \right)\frac{1 + v_{f}}{1 + v_{s}}T} + {\begin{bmatrix}{{\left( {1 + v_{s} - {2\frac{1 + v_{f}}{1 + v_{s}}}} \right)\alpha_{s}} +} \\{\frac{1 + v_{f}}{1 + v_{s}}\left( {1 - v_{s}} \right)\alpha_{f}}\end{bmatrix}\left( {T - {\overset{\_}{T}(R)}} \right)} + {\frac{2}{3 + v_{s}}\left\{ {{\left( {1 - v_{s}} \right)\begin{bmatrix}{{\left( {1 + v_{s}} \right)\alpha_{s}} -} \\{\left( {1 + v_{f}} \right)\alpha_{f}}\end{bmatrix}} + {4\left( {v_{f} - v_{s}} \right)\alpha_{s}}} \right\} \times {\sum\limits_{n = 1}^{\infty}{\frac{\left( {n + 1} \right)r^{n}}{R^{{2n} + 2}}\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{1 - n}T_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}}} \right\rangle}} & (19)\end{matrix}$It is noted that for purely axisymmetric (radially symmetric) variationsEquation (19) reduces to Eq. (7). This is true since there is nocircumferential variation and as a result, the parameters T_(c) ^((n))and T_(s) ^((n)) both vanish for every integer n (see eq. (18)). For themost general case, the coefficients of the infinite sum involveintegrals over r (from 0 to R) and over θ (from 0 to 2π) arising fromthe definition, of T_(c) ^((n)) and T_(s) ^((n)). This demonstrates thenon-local nature of the result.

The general relation between the difference of direct curvatures and Tare given by the following: $\begin{matrix}\left. {{K_{rr} + K_{\theta\theta}} = {{{\frac{6E_{f}{h_{f}\left( {1 - v_{s}^{2}} \right)}}{E_{s}{h_{s}^{2}\left( {1 - v_{f}^{2}} \right)}}\begin{bmatrix}{{\left( {1 + v_{s}} \right)\alpha_{s}} -} \\{\left( {1 + v_{f}} \right)\alpha_{f}}\end{bmatrix}}\begin{Bmatrix}\begin{matrix}{{\left\langle {T - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad T^{(0)}{\mathbb{d}\eta}}}}} \right\rangle -}\quad} \\{{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\begin{bmatrix}{{{\cos\quad n\quad\theta\quad{\int_{0}^{r}\quad{\eta^{1 + n}\quad T_{c}^{(n)}\quad{\mathbb{d}\eta}}}} +}\quad} \\{\sin\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{1 + n}\quad T_{s}^{(n)}\quad{\mathbb{d}\eta}}}}\end{bmatrix}}} -}\end{matrix} \\{\sum\limits_{n = 1}^{\infty}{\left( {n - 1} \right){r^{n - 2}\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{r}^{R}{\eta^{1 - n}T_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{r}^{R}{\eta^{1 - n}T_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}}\end{Bmatrix}} + {\frac{6E_{f}{h_{f}\left( {1 - v_{s}^{2}} \right)}}{E_{s}{h_{s}^{2}\left( {1 - v_{f}^{2}} \right)}}\frac{1}{3 + v_{s}}\begin{Bmatrix}{{\left( {1 - v_{\quad s}} \right)\begin{bmatrix}{{\left( {1 + v_{s}} \right)\quad\alpha_{s}} -} \\{\left( {1 + v_{f}} \right)\quad\alpha_{f}}\end{bmatrix}} +} \\{4\left( {v_{f} - v_{s}} \right)\alpha_{\quad s}}\end{Bmatrix}*{\sum\limits_{n = 1}^{\infty}{{\frac{n + 1}{R^{n + 2}}\begin{bmatrix}{{n\left( \frac{r}{R} \right)}^{n} -} \\{\left( {n - 1} \right)\left( \frac{r}{R} \right)^{n - 2}}\end{bmatrix}}\begin{bmatrix}{{{\cos\quad n\quad\theta\quad{\int_{0}^{R}{\eta^{1 + n}\quad T_{c}^{(n)}{\mathbb{d}\eta}}}} +}\quad} \\{\sin\quad n\quad\theta\quad{\int_{o}^{R}{\eta^{1 + n}\quad T_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}}}} \right\} & (20)\end{matrix}$The line integrals are evaluated over circular discs or radius of r andover annular regions ranging from r to R. Such integrals are of thetype: ∫_(r)^(R)η^(1 ± n)(…)𝕕η  or  ∫_(o)^(R)η^(1 ± n)(…)𝕕η.It is also noted that (K_(rr)−K_(θθ)) vanishes for the special case ofthe uniform temperature (as expected in the restricted case by Stoney).

In the most general case treated here, the twist curvatures K_(rθ) doesnot vanish and is given by: $\begin{matrix}{K_{r\quad\theta} = {{{\frac{3E_{f}{h_{f}\left( {1 - v_{s}^{2}} \right)}}{E_{s}{h_{s}^{2}\left( {1 - v_{f}^{2}} \right)}}\begin{bmatrix}{{\left( {1 + v_{s}} \right)\alpha_{s}} -} \\{\left( {1 + v_{f}} \right)\alpha_{f}}\end{bmatrix}}\begin{Bmatrix}{{- {\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\begin{bmatrix}{{\sin\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{1 + n}T_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{0}^{r}{\eta^{1 + n}T_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}} +} \\{\sum\limits_{n = 1}^{\quad\infty}{\left( {n - 1} \right){r^{\quad{n\quad - \quad 2}}\begin{bmatrix}{{{\sin\quad n\quad\theta\quad\int_{r}^{R}\quad\eta^{1 - n}\quad T_{\quad c}^{(n)}\quad{\mathbb{d}\eta}}\quad -}\quad} \\{\cos\quad n\quad\theta\quad\int_{r}^{R}\quad\eta^{1 - n}\quad T_{\quad s}^{(n)}\quad{\mathbb{d}\eta}}\end{bmatrix}}}}\end{Bmatrix}} - {\frac{3E_{f}{h_{f}\left( {1 - v_{s}^{2}} \right)}}{E_{s}{h_{s}^{2}\left( {1 - v_{f}^{2}} \right)}}\frac{1}{3 + v_{s}}\begin{Bmatrix}{{\left( {1 - v_{s}} \right)\begin{bmatrix}{{\left( {1 + v_{\quad s}} \right)\alpha_{\quad s}} -} \\{\left( {1 + v_{f}} \right)\alpha_{f}}\end{bmatrix}} +} \\{4\left( {v_{f} - v_{s}} \right)\alpha_{s}}\end{Bmatrix} \times {\sum\limits_{n = 1}^{\infty}{{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\left( \frac{r}{R} \right)}^{n} - {\left( {n + 1} \right)\left( \frac{r}{R} \right)^{n - 2}}} \right\rbrack}\begin{bmatrix}{{{\sin\quad n\quad\theta\quad{\int_{0}^{R}{\eta^{1 + n}T_{c}^{(n)}\quad{\mathbb{d}\eta}}}} -}\quad} \\{\cos\quad n\quad\theta\quad\int_{0}^{R}\quad\eta^{1 + n}\quad T_{s}^{(n)}\quad{\mathbb{d}\eta}}\end{bmatrix}}}}}} & (21)\end{matrix}$The right hand side of Eq. (21) vanishes for a constant temperaturedistribution and for a temperature distribution with a radial onlyvariation. For both of these special cases, Eq. (18) predicts T_(c)^((n))=T_(s) ^((n))=0 for every positive integer n=1, 2, 3 . . . .

Relations in Eqs. (19)-(21) can be reduced to the previously presentedformulae for the special case where a layered plate structure isaxisymmetric (i.e., radially symmetric) so that the temperature T is nota function of θ. If T is a constant and does not change spatially, Eqs.(19)-(21) can be further reduced to the predictions from the highlyrestrictive Stoney formula. Additional relations can be derived from theabove equations. For example, adding Eqs. (19) and (20) and dividing theresult by two provides a relation between K_(rr) and T(r,θ,t). Asanother example, subtracting Eq. (20) from Eq. (19) and dividing theresult by two provides K_(θθ)(r,θ,t) as a function of T(r,θ,t).

All of these relations are clearly non-local in nature. Hence, thetemperature at a particular location r,θ influences the curvaturecomponents in all other locations on the wafer. When the spatialvariation is small, e.g., the spatial temperature gradients are small,the curvature components at one location may be mainly affected by thelocal temperature as predicted by the Stoney formula. In many practicalsituations in semiconductor processing and other fabrications of otherlayered plate structures, the effects of spatial variations cannot beignored and the non-local contributions are significant. Therefore, theStoney formula may become inadequate.

The following section discusses a way of inverting Eq. (19) to provide amethod for inferring arbitrary spatial temperature distributions fromthe measurement of the system's first curvature invariant. Forconvenience, the following coefficients are defined: $\begin{matrix}{{C_{n} = {\frac{1}{\pi\quad R^{2}}\underset{A}{\int\int}\left( {K_{rr} + K_{\theta\theta}} \right)\left( \frac{\eta}{R} \right)^{n}\cos\quad n\quad\varphi\quad{\mathbb{d}A}}}{S_{n} = {\frac{1}{\pi\quad R^{2}}\underset{A}{\int\int}\left( {K_{rr} + K_{\theta\theta}} \right)\left( \frac{\eta}{R} \right)^{n}\sin\quad n\quad\varphi\quad{\mathbb{d}A}}}} & (22)\end{matrix}$where A is the area of the wafer and is equal to πR² for a circularwafer with a radius of R. For n=0, the coefficient C₀ is given by$\begin{matrix}{C_{0} = {{\frac{1}{\pi\quad R^{2}}{\int{\int{\left( {K_{rr} + K_{\theta\theta}} \right){\mathbb{d}A}}}}} = \overset{\_}{K_{rr} + K_{\theta\theta}}}} & (23)\end{matrix}$which is the area averaged value of K_(rr)+K_(θθ)(r,θ,t). For thisparticular case, the coefficient S₀=0.

Taking the area average of both sides of Eq. (19) furnishes a relationbetween K_(rr)+K_(θθ) and T as follows: $\begin{matrix}{\overset{\_}{T} = {\frac{E_{s}{h_{s}^{2}\left( {1 - v_{f}} \right)}}{12E_{f}{h_{f}\left( {1 - v_{s}} \right)}}\frac{\overset{\_}{K_{rr} + K_{\theta\theta}}}{\alpha_{s} - \alpha_{f}}}} & (24)\end{matrix}$From Eqs. (19) and (24), the relation between T(r,θ,t) andK_(rr)+K_(θθ)(r,θ,t) can be expressed as follows: $\begin{matrix}{{T\left( {r,\theta,t} \right)} = {{\frac{E_{s}{h_{s}^{2}\left( {1 - v_{f}^{2}} \right)}}{6E_{f}{h_{f}\left( {1 - v_{s}^{2}} \right)}}\frac{1}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}}\begin{Bmatrix}{K_{rr} + K_{\theta\theta} - \overset{\_}{K_{rr} + K_{\theta\theta}} -} \\{\frac{{\left( {1 - v_{s}} \right)\begin{bmatrix}{{\left( {1 + v_{\quad s}} \right)\alpha_{\quad s}} -} \\{\left( {1 + v_{f}} \right)\alpha_{f}}\end{bmatrix}} + {4\left( {v_{f} - v_{s}} \right)\alpha_{s}}}{2\left( {1 + v_{f}} \right)\left( {\alpha_{s} - \alpha_{f}} \right)} \times} \\{\sum\limits_{n = 1}^{\quad\infty}{\left( {n + 1} \right){\left( \frac{r}{R} \right)^{n}\left\lbrack {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right\rbrack}}}\end{Bmatrix}} + {\frac{E_{s}{h_{s}^{2}\left( {1 - v_{f}} \right)}}{12E_{f}{h_{f}\left( {1 - v_{s}} \right)}}\frac{\overset{\_}{K_{rr} + K_{\theta\theta}}}{\alpha_{s} - \alpha_{f}}}}} & (25)\end{matrix}$where C_(r) and S_(f) are given in Eq. (22) in terms of area integralsinvolving (K_(rr)+K_(θθ))(r,θ,t). Eq. (25) allows for the inference ofT(r,θ,t) once K_(rr)+K_(θθ)=K₁+K₂=K₁₁+K₂₂ is measured in full field. Afull field interferometric method, e.g. any shearing interferometer suchas CGS or any other type of optical topography mapping device, can beused to measure K_(rr) and K_(θθ).

Equation (25) reduces to the equivalent axisymmetric result for radiallysymmetric temperature distributions (see Eqs. (8) and (9)).

The following sections provide generalized relations between filmstresses and temperatures increase for layered plate structures witharbitrary spatial variations in r and θ.

We further define the following notations:

σ_(rr)(r,θ,t): direct stress in the radial direction;

σ_(θθ)(r,θ,t): direct hoop (or circumferential) stress;

σ_(rθ)(r,θ,t)=σ_(θr)(r,θ,t): in-plane shear stress in the plane of film(polar coordinates);

σ₃₃(r,θ,t)=0: the out of plane direct stress vanishes because of theassumption that each film on the substrate is thin relative to thesubstrate;

τ_(r)=σ_(3r)(r,θ,t)=σ_(r3) (r,θ,t): shear stress, along the radialdirection, acting on film at the interface between the film and thesubstrate; and

τ_(θ)=σ_(3θ)(r,θ,t)=σ_(θ3)(r,θ,t): shear stress, along the hoopdirection, acting on film at the interface between the film and thesubstrate.

FIG. 1B illustrates various stresses defined above.

Based on the thermoelastic plate theory and small deformationkinematics, the relation between the above-defined stress components anda spatial-varying temperature profile on the film-substrate system canbe expressed as follows. First the sum of the diagonal stress tensorelements, which is an invariant, is given by: $\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{f}}{1 - v_{f}}\begin{Bmatrix}{{2\left( {\alpha_{s} - \alpha_{f}} \right)T} - {\left( {1 - v_{s}} \right)\alpha_{s}\left( {T - \overset{\_}{T}} \right)} + {2\left( {1 - v_{\quad s}} \right)\alpha_{\quad s}{\sum\limits_{n\quad = \quad 1}^{\quad\infty}{\frac{n\quad + \quad 1}{\quad R^{\quad{{2\quad n}\quad + \quad 2}}}r^{\eta}}}}} \\\left\lbrack {{\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{\quad{1 + n}}T_{c}^{(n)}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{\quad{1 + n}}T_{s}^{(n)}{\mathbb{d}\eta}}}}} \right\rbrack\end{Bmatrix}}} & (26)\end{matrix}$where the spatial average temperature over the entire substrate disk is$\overset{\_}{T} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{T{\mathbb{d}A}}}}}$Notably, the first term in Eq. (26) is Stoney-like and reflects thelocal contribution; the second term is similar in structure to theaxisymmetric case and reflects the first-order non-local contribution(see, Eq. (11)). In addition, the third term in Eq. (26) is also highlynon-local and vanishes only if the temperature increase has nodependence on θ.

Second, the difference between the diagonal stress tensor elements isgiven by: $\begin{matrix}{{{\sigma_{rr} - \sigma_{\theta\theta}} = {\frac{{E_{f}\left( {1 + v_{s}} \right)}\alpha_{s}}{1 + v_{f}}\begin{Bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}{T - {\frac{2}{r^{2}}{\int_{0}^{r}{{\eta\Delta}\quad T^{(0)}{\mathbb{d}\eta}}}} -} \\{{\sum\limits_{n = 1}^{\quad\infty}{\frac{n + 1}{\quad r^{\quad{n + 2}}}\begin{bmatrix}{{{\cos\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{n + 1}\quad T_{c}^{(n)}\quad{\mathbb{d}\eta}}}} +}\quad} \\{\sin\quad n\quad\theta\quad\int_{0}^{\quad r}\quad\eta^{n + 1}\quad T_{s}^{(n)}\quad{\mathbb{d}\eta}}\end{bmatrix}}} -}\end{matrix} \\{{\sum\limits_{n = 1}^{\quad\infty}{\left( {n - 1} \right){r^{\quad{n - 2}}\begin{bmatrix}{\cos\quad n\quad\theta\quad\int_{r}^{R}\quad\eta^{1 - \quad n}\quad T_{c}^{(n)}\quad{\mathbb{d}{\eta++}}} \\{\sin\quad n\quad\theta\quad\int_{r}^{R}\quad\eta^{1 - n}\quad T_{s}^{(n)}\quad{\mathbb{d}\eta}}\end{bmatrix}}}} -}\end{matrix} \\{\sum\limits_{n = 1}^{\quad\infty}{\frac{n + 1}{\quad R^{n + 2}}\left\lbrack {{n\left( \quad\frac{r}{\quad R} \right)}^{n} - {\left( {n - 1} \right)\left( \quad\frac{r}{\quad R} \right)^{n - 2}}} \right\rbrack}}\end{matrix} \\\begin{bmatrix}{{{\cos\quad n\quad\theta\quad{\int_{0}^{R}\quad{\eta^{1 + n}\quad T_{c}^{(n)}\quad{\mathbb{d}\eta}}}} +}\quad} \\{\sin\quad n\quad\theta\quad\int_{0}^{R}\quad\eta^{1 + n}\quad T_{s}^{(n)}\quad{\mathbb{d}\eta}}\end{bmatrix}\end{Bmatrix}}}\quad} & (27)\end{matrix}$where the first term on the right hand side of Eq. (27) is similar instructure to the axisymmetric case (Eq. (12)) and the following termswith infinite series vanish when the temperature T is axisymmetric. Bothterms in Eq. (27) vanish for T(r, θ, t)=Tc=constant so that Eq. (27)reduces to the Stoney formula.

The in-plane shear stress component is given by $\begin{matrix}{\sigma_{r\quad\theta} = {\frac{{E_{f}\left( {1 + v_{s}} \right)}\alpha_{s}}{2\left( {1 + v_{f}} \right)}\begin{Bmatrix}\begin{matrix}\begin{matrix}{{- {\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\begin{bmatrix}{{\sin\quad n\quad\theta{\int_{0}^{r}{\eta^{1 + n}T_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{0}^{r}{\eta^{1 + n}T_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}} +} \\{{\sum\limits_{n = 1}^{\infty}{\left( {n - 1} \right){r^{n\quad - \quad 2}\begin{bmatrix}{{\sin\quad n\quad\theta\quad{\int_{r}^{R}{{\quad\eta^{1 - \quad n}}\quad T_{c}^{(n)}\quad{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta\quad{\int_{r}^{R}{{\quad\eta^{1 - \quad n}}\quad T_{s}^{(n)}\quad{\mathbb{d}\eta}}}}\end{bmatrix}}}} +}\end{matrix} \\{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\left( \frac{r}{R} \right)}^{n} - {\left( {n - 1} \right)\left( \frac{r}{R} \right)^{n - 2}}} \right\rbrack}}\end{matrix} \\\left\lbrack {{\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{c}^{(n)}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{s}^{(n)}{\mathbb{d}\eta}}}}} \right\rbrack\end{Bmatrix}}} & (28)\end{matrix}$shear stress component is entirely caused by non-local contributions.For both radially symmetric and constant T, the stress component σ_(rθ)above becomes zero.

The interfacial shear stress components along the radial direction andalong the hoop or polar direction are $\begin{matrix}\begin{matrix}{\tau_{r} = \sigma_{3r}} \\{= {\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\begin{Bmatrix}{{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack\frac{\partial T}{\partial r}} +} \\{2\left( {v_{f} - v_{s}} \right)\alpha_{s}{\sum\limits_{n = 1}^{\infty}{{n\left( {n + 1} \right)}\frac{r^{n - 1}}{R^{{2n} + 2}}}}} \\\left\lbrack {{\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{c}^{(n)}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{s}^{(n)}{\mathbb{d}\eta}}}}} \right\rbrack\end{Bmatrix}}}\end{matrix} & (29) \\\begin{matrix}{\tau_{\theta} = \sigma_{3\theta}} \\{= {\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\begin{Bmatrix}{{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack\frac{1}{r}\frac{\partial T}{\partial\theta}} -} \\{2\left( {v_{f} - v_{s}} \right)\alpha_{s}{\sum\limits_{n = 1}^{\infty}{{n\left( {n + 1} \right)}\frac{r^{n - 1}}{R^{{2n} + 2}}}}} \\\left\lbrack {{\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{c}^{(n)}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{1 + n}T_{s}^{(n)}{\mathbb{d}\eta}}}}} \right\rbrack\end{Bmatrix}}}\end{matrix} & (30)\end{matrix}$Hence, a spatially non-uniform T is required to generate interfacialshear stresses. For axisymmetric temperatures i layered platestructures, Eq. (29) reduces to Eq. (13) while Eq. (30) predicts a zerovalue for τ_(θ). If, in addition, T is spatially uniform (i.e.,$\left. {\frac{\partial T}{\partial\theta} = {\frac{\partial T}{\partial r} = 0}} \right),$then these shear stresses vanish everywhere as is the case for Stoney'srestrictive approach.

Elimination of temperature from the curvature/temperature and thestress/temperature relations (Eqs.(19)-(21) and (26)-(30) respectively)provides the connection between film stresses and system curvatures forthe most general case of arbitrary film stress variations. The resultsprovide a simple analytical framework for the accurate and fastmeasurement of spatially and temporally varying stress fields throughfull field measurements of the complete curvature tensor. The resultsshow that the classical Stoney approach may be inadequate, and incertain cases may be highly inaccurate since it has been derived on thebasis of the assumption of spatial uniformity. The axisymmetric case isrecovered as a special case.

Recall the definitions of C_(n) and S_(n) which are integrals in Eq.(22) and involve the first curvature invariant (I=K_(rr)+K_(θθ)/2). Inaddition, recall that S₀=0, while C₀ is the area average of(K_(rr)+K_(θθ)). Based on these, the relations between the in-plane filmstresses σ_(rr), σ_(θθ) and σ_(rθ) and the system curvatures K_(rr),K_(θθ) and K_(rθ) can be expressed as follows: $\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{s}h_{s}^{2}}{6{h_{f}\left( {1 - v_{s}} \right)}}\begin{Bmatrix}\begin{matrix}{K_{rr} + K_{\theta\theta} + \left\lbrack {\frac{1 - v_{s}}{1 + v_{s}} - \frac{\left( {1 - v_{f}} \right)\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}}} \right\rbrack} \\{\left( {K_{rr} + K_{\theta\theta} - \overset{\_}{K_{rr} + K_{\theta\theta}}} \right) + {\sum\limits_{n = 1}^{\infty}{\left( {n + 1} \right)\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)}}}\end{matrix} \\{\left( \frac{r}{R} \right)^{n}\left\lbrack {\frac{2\left( {1 - v_{f}} \right)\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} - \frac{1 - v_{s}}{1 + v_{s}}} \right\rbrack}\end{Bmatrix}}} & (31) \\{{\sigma_{rr} - \sigma_{\theta\theta}} = {\frac{E_{s}{h_{s}^{2}\left( {1 - v_{f}} \right)}}{6{h_{f}\left( {1 - v_{s}} \right)}}\frac{\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \times \begin{Bmatrix}{K_{rr} - K_{\theta\theta} - {\sum\limits_{n = 1}^{\infty}\left( {n + 1} \right)}} \\{\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)\left\lbrack {{n\left( \quad\frac{r}{\quad R} \right)}^{n} - {\left( {n - 1} \right)\left( \quad\frac{r}{\quad R} \right)^{n - 2}}} \right\rbrack}\end{Bmatrix}}} & (32) \\{\sigma_{r\quad\theta} = {\frac{E_{s}{h_{s}^{2}\left( {1 - v_{f}} \right)}}{6{h_{f}\left( {1 - v_{s}} \right)}}\frac{\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \times \begin{Bmatrix}{K_{r\quad\theta} + {\frac{1}{2}{\sum\limits_{n = 1}^{\infty}{\left( {n + 1} \right)\left( {{C_{n}\sin\quad n\quad\theta} - {S_{n}\cos\quad n\quad\theta}} \right)}}}} \\\left\lbrack {{n\left( \quad\frac{r}{\quad R} \right)^{n}} - {\left( {n - 1} \right)\left( \quad\frac{r}{\quad R} \right)^{n\quad - \quad 2}}} \right\rbrack\end{Bmatrix}}} & (33)\end{matrix}$where the first terms of Eqs. (31) and (32) are identical in structureto the axisymmetric case described in Eqs. (14) and (15). In addition,the subsequent terms with infinite series of Eqs. (31) and (32) involvearea integrals whose integrands are weighted contributions fromK_(rr)+K_(θθ) only. Eq. (33) also has a structure similar to Eqs. (31)and (32) but both sides of Eq. (33) become identically zero for theaxisymmetric case.

The relation between interfacial shears and curvatures are given by:$\begin{matrix}\begin{matrix}{\tau_{r} = \sigma_{3r}} \\{= {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\begin{Bmatrix}{{\frac{\partial}{\partial r}\left( {K_{\quad{rr}} + K_{\quad{\theta\theta}}} \right)} - {\frac{1 - v_{\quad s}}{\quad{2\quad R}}{\sum\limits_{n = 1}^{\quad\infty}{n\left( {n + 1} \right)}}}} \\{\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)\left( \frac{r}{R} \right)^{n - 1}}\end{Bmatrix}}}\end{matrix} & (34) \\\begin{matrix}{\tau_{\theta} = \sigma_{3\theta}} \\{= {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\begin{Bmatrix}{{\frac{1}{r}\frac{\partial}{\partial\theta}\left( {K_{\quad{rr}} + K_{\quad{\theta\theta}}} \right)} + {\frac{1 - v_{\quad s}}{\quad{2\quad R}}{\sum\limits_{n = 1}^{\quad\infty}{n\left( {n + 1} \right)}}}} \\{\left( {{C_{n}\sin\quad n\quad\theta} - {S_{n}\cos\quad n\quad\theta}} \right)\left( \frac{r}{R} \right)^{n - 1}}\end{Bmatrix}}}\end{matrix} & (35)\end{matrix}$The shear stress components acting in the interface between the film andthe substrate depend on the gradient of K_(rr)+K_(θθ) and on non-localcontributions through the area integrals C_(n) and S_(n) which involveK_(rr)+K_(θθ). The pure dependence of these expressions on the firstinvariant of the curvature is noted here.

When the first invariant is axisymmetric, then C_(n)=S_(n)=0 and:$\begin{matrix}{{\tau_{r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\frac{\partial}{\partial r}\left( {K_{rr} + K_{\theta\theta}} \right)}}{\tau_{\theta} = {{\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\frac{1}{r}\frac{\partial}{\partial\theta}\left( {K_{rr} + K_{\theta\theta}} \right)} = 0}}} & (36)\end{matrix}$which is consistent with the predictions of the axisymmetric specialcase discussed above.

The infinite series representation in Equations (31)-(35) can berigorously substituted by a more complex to evaluate but more compactnon-local integral form. The two forms are mathematically equivalent.Three examples given here: $\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{s}h_{s}^{2}}{6{h_{f}\left( {1 - v_{s}} \right)}} \times \begin{Bmatrix}{\left( {K_{rr} + K_{\theta\theta}} \right) + \left\lbrack {\frac{1 - v_{s}}{1 + v_{s}} - \frac{\left( {1 - v_{f}} \right)\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}}} \right\rbrack} \\{\left( {K_{rr} + K_{\theta\theta} - \overset{\_}{K_{rr} + K_{\theta\theta}}} \right) +} \\{{\left\lbrack {\frac{2\left( {1 - v_{f}} \right)\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} - \frac{1 - v_{s}}{1 + v_{s}}} \right\rbrack \cdot \frac{r}{\pi\quad R^{3}}} \times} \\{\underset{A}{\int\int}\left( {K_{rr} + K_{\theta\theta}} \right)\frac{\eta}{R}} \\{\frac{\begin{matrix}{{2\cos{\left( {\varphi - \theta} \right)\left\lbrack {1 + {2\frac{\quad{\eta^{\quad 2}\quad r^{\quad 2}}}{\quad R^{\quad 4}}}} \right\rbrack}} -} \\{{\frac{\eta\quad r}{\quad R^{\quad 2}}{\cos\left( {{2\varphi} - {2\theta}} \right)}} - {\frac{\eta\quad r}{\quad R^{\quad 2}}\left\lbrack {4 + \frac{\quad{\eta^{\quad 2}\quad r^{\quad 2}}}{\quad R^{\quad 4}}} \right\rbrack}}\end{matrix}}{\left\lbrack {1 - {2\frac{\eta\quad r}{R^{2}}{\cos\left( {\varphi - \theta} \right)}} + \frac{\eta^{\quad 2}r^{\quad 2}}{R^{4}}} \right\rbrack^{2}}{\mathbb{d}A_{\eta\varphi}}}\end{Bmatrix}}} & (37)\end{matrix}$where the integration variables are η and φ. $\begin{matrix}{\tau_{\theta} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\left\langle {{\frac{1}{r}\frac{\partial}{\partial\theta}\left( {K_{rr} + K_{\theta\theta}} \right)} - {\frac{\left( {1 - v_{s}} \right)}{\pi\quad R^{3}}{\int{\int\limits_{A}{\left( {K_{rr} + K_{\theta\theta}} \right){\frac{\eta}{R} \cdot \frac{{\left( {1 - {3\frac{\eta^{2}r^{2}}{R^{4}}}} \right){\sin\left( {\varphi - \theta} \right)}} + {\frac{\eta^{3}r^{3}}{R^{6}}{\sin\left( {{2\varphi} - {2\theta}} \right)}}}{\left\lbrack {1 - {2\frac{\eta\quad r}{R^{2}}{\cos\left( {\varphi - \theta} \right)}} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}}{\mathbb{d}A_{\eta\varphi}}}}}}} \right.}} & (38) \\{\tau_{r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\begin{matrix}\left\langle {{\frac{\partial}{\partial r}\left( {K_{rr} + K_{\theta\theta}} \right)} - {\frac{\left( {1 - v_{s}} \right)}{\pi\quad R^{3}}{\int{\int\limits_{A}{\left( {K_{rr} + K_{\theta\theta}} \right){\frac{\eta}{R} \cdot}}}}}} \right. \\\left. {\frac{{\left( {1 - {3\frac{\eta^{2}r^{2}}{R^{4}}}} \right){\cos\left( {\varphi - \theta} \right)}} - \left\lbrack {{3\frac{\eta\quad r}{R^{2}}} + {\frac{\eta^{3}r^{3}}{R^{6}}{\cos\left( {{2\varphi} - {2\theta}} \right)}}} \right\rbrack}{\left\lbrack {1 - {2\frac{\eta\quad r}{R^{2}}{\cos\left( {\varphi - \theta} \right)}} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A_{\eta\varphi}}} \right\rangle\end{matrix}}} & (39)\end{matrix}$

Equations (31) to (39) represent the general formulations which relatestress and curvature components and their invariants. The results holdfor arbitrarily varying deformation and stresses. They are referred tohere as the H.R. stress-curvature relations and can appropriately reduceto the axisymmetric formulation for radial symmetry and to therestrictive Stoney relation for spatially uniform states.

The H.R. relations are clearly non-local despite the fact that they havebeen obtained through the use of linearized kinematics (small straintheory) and a linear, elastic, constitutive law. When the H.R. relationsare presented in the form which involves an infinite series, theirnon-local nature is demonstrated through the area integrals used tocalculate the coefficients C_(n) and S_(n). The integrants of theseintegrals involve expression of the form:$K_{rr} + {{{K_{\theta\theta}\left( {\eta,\varphi} \right)} \cdot \left( \frac{\eta}{R} \right)^{n}}\begin{Bmatrix}{\cos\quad n\quad\varphi} \\{\sin\quad n\quad\varphi}\end{Bmatrix}}$where η,φ are the two spatial integration variables. As n increases,(η/R)^(n) drops drastically for most points on the wafer since η≦R,making C_(n) and S_(n) decreasingly smaller.

As a consequence of this, we expect that only a few limited terms, e.g.,three to five terms, in the series (depending on the severity of thenon-uniformity with respect to the angular parameter θ in K_(rr)+K_(θθ))will be needed for most cases encountered in practice. We expect thatthis will allow for the early truncation of the series making theprocess of evaluation straight forward and quick. Of significance hereis that these non-local contributions demonstrate themselves onlythrough the combination K_(rr)+K_(θθ) which is equal to twice the firstcurvature invariant I. Indeed all the area integrals involved in theevaluation of every single stress component involve the first curvatureinvariant and no other contribution from individual curvature componentsor other invariants.

The integral forms involved in equations (37)-(39) are compact in formand physically revealing. Stresses at each point (r,θ) on the film aredependent on:

-   -   1. The local value of individual curvature components. This is        referred to here as the Stoney-like contribution.    -   2. The deviation of the local value of K_(rr)+K_(θθ) from its        area-averaged value K_(rr)+K_(θθ) . This is a primitive form of        non-locality and is referred to here as the axisymmetric-like        contribution.    -   3. The value of K_(rr)+K_(θθ) from all other points on the        wafer. This is the non-local contribution and can become        important when the first curvature invariant has a non-uniform θ        distribution.

The above formulation can trivially be extended to cases where the filmdeposited on the substrate is replaced by a composite film structurewhich is also mechanically and thermally isotropic. In such a case thethermal and mechanical properties α_(f), E_(f), and ν_(f) should bereplaced by their composite values α_(f) ^(c),E_(f) ^(c) and v_(f) ^(c)computed by a simple rule of mixtures involving the volume fractions andproperties of the individual constituents. In this case the computedstresses simple are the averages of the stresses acting over thethickness of the composite film structure.

The above techniques for analyzing film stresses and curvatures due tospatially non-uniform temperature distributions in both spatiallyaxisymmetric and arbitrary profiles remove certain restrictions in theclassic Stoney formulation and provide relations between the filmstresses and the temperature and the plate curvatures and thetemperature with both Stoney-like local contributions and non-localcontributions that are missing from Stoney's formulation. Thetemperature analyses are based on the assumption that both the substrateand the film at each location are at the same temperature and thus thedriving force for the system curvature is the temperature distribution.As such, the analyses do not include the effects to the curvatures andstresses from other sources, such as the misfit strain between the filmand the underlying substrate. Due to differences in material propertiesbetween the film and the substrate and due to impacts of variousprocesses applied during the fabrication, the misfit strain between thefilm and the underlying substrate can be a significant source to thestresses and curvatures. Hence, it is desirable to include the misfitstrain in the analysis to provide tools for relating film stresses andsystem curvatures to the misfit strain distribution, and for providing arelation between the film stresses and the system curvatures forspatially varying, in-plane isotropic misfit strain distributions. Suchtools can be used to provide accurate experimental inference ofaccumulated film stress from full-field curvature measurements which maytake place following various processing steps for on-line monitoring,fabrication process analysis, wafer examination and other applications.

As discussed above, many other methodologies used for the inference ofthin film stress through curvature measurements are strictly restrictedto stress and curvature states which are assumed to remain uniform overthe entire film/substrate system. By considering a circularthin-film/substrate system subject to arbitrarily non-uniform misfitstrain distributions, we derive relations between the film stresses andthe misfit strain, and between the plate system's curvatures and themisfit strain. These relations feature a “local” part which involves adirect dependence of the stress or curvature components on the misfitstrain at the same point, and a “non-local” part which reflects theeffect of misfit strain of other points on the location of scrutiny.Most notably, we also derive relations between components of the filmstress and those of system curvatures which allow for the experimentalinference of such stresses from full-field curvature measurements in thepresence of arbitrary non-uniformities. These relations also feature a“non-local” dependence on curvatures making full-field measurements ofcurvature a necessity for the correct inference of stress. Finally, itis shown that the interfacial shear tractions between the film and thesubstrate are related to the gradients of the first curvature invariantand can also be inferred experimentally.

Thin film structures on substrates are often made from a multiplicity offabrication and processing steps (e.g., sequential film deposition,thermal anneal and etch steps) and often experience stresses caused byeach of these steps. Examples of known phenomena and processes thatbuild up stresses in thin films include, but are not limited to, latticemismatch, chemical reaction, doping by e.g., diffusion or implantation,rapid deposition by evaporation or sputtering and of course thermaltreatment (e.g., various thermal anneal steps). The film stress build upassociated with each of these steps often produces undesirable damagethat may be detrimental to the manufacturing process because of itscumulative effect on process “yield.” Known problems associated tothermal excursions, in particular, include stress-induced film crackingand film /substrate delamination resulting during uncontrolled wafercooling which follows the many anneal steps.

The intimate relation between stress-induced failures and process yieldloss makes the identification of the origins of stress build-up, theaccurate measurement and analysis of stresses, and the acquisition ofinformation on the spatial distribution of stresses a crucial step indesigning and controlling processing steps and in ultimately improvingreliability and manufacturing yield. Stress changes in thin filmsfollowing discrete process steps or occurring during thermal excursionsmay be calculated in principle from changes in the film/substratesystems curvatures or “bow” based on analytical correlations betweensuch quantities.

The techniques disclosed in this application remove the two restrictiveassumptions of the Stoney analysis relating to spatial uniformity andequi-biaxiality. Previous sections of this application provide analyseson the cases of thin film/substrate systems subject to a non-uniform,axisymmetric temperature distribution T(r) and an “arbitrary”temperature distribution T(r,θ). The results of this generalization aresubstantially much more complicated than those of the axisymmetric caseT(r) but have a very similar structure. As perhaps expected they can bedecomposed to a “local” or “Stoney-like” part, and a non-local part. Thefirst term of the non-local part is identical in structure to that ofthe axisymmetric prediction while the rest is given in terms of aninfinite series of terms of diminishing strength.

An example of non-axisymmetric curvature distribution is given in FIG.1C. FIGS. 1C(a)-(c) show the x-direction, y-direction and mixedcurvature distributions of a large, industrial wafer composed of a low-kdielectric film deposited on a Si substrate. The curvatures have beenobtained by means of CGS interferometry, a technique capable ofperforming full-field, real-time measurements of all three independentCartesian components (κ_(xx), κ_(yy) and κ_(xy)=κ_(xy)) of the curvaturetensor over the entire wafer. The non-axisymmetry of these maps as wellas the clear presence of large scale curvature non-uniformities providesstrong motivation for this study. This non-uniformity is in clearviolation of Stoney's 6^(th) assumption. Furthermore the two maps inFIGS. 1C(a) and 1C(b) are clearly different. This is also in clearviolation of Stoney's 5^(th) assumption which requires equi-biaxialityof curvature.

The generalization of the axisymmetric misfit strain ε_(m)(r) to anarbitrarily varying misfit strain ε_(m)(r,θ) is the subject of thepresent investigation. Indeed, the main purpose of the present paper isto remove the two restrictive assumptions of the Stoney analysisrelating to stress and curvature spatial uniformity and to in-planeisotropic equi-biaxiality for the general case of a thin film/substratesystem subject to an arbitrarily varying misfit strain distributionε_(m)(r,θ) whose presence can create an arbitrary stress and curvaturefield as well as arbitrarily large stress and curvature gradients. Itshould be noted that this misfit strain although spatially varying it islocally assumed to be in-plane isotropic. This does not imply that thestress state is also in-plane isotropic.

Although many important features of the two solutions, corresponding toT(r,θ) or ε_(m)(r,θ) non-uniformities are expected to be similar, somefundamental differences between these two situations are alsoanticipated. For T(r,θ), the driving force for system curvature is thetemperature distribution while for ε_(m)(r,θ) the driving force is isthe misfit strain between film and substrate. In T(r,θ), both the filmand the substrate are each subjected to T(r,θ) and even if not bondedthey independently develop non-uniform deformation and stress states.These states need to be further reconsidered due to eventualfilm/substrate bonding (continuity of displacements across theinterface). In ε_(m)(r,θ), however, the film misfit strain induces thesystem deformations and the film and substrate stress. Accordingly, inthe limit of zero film thickness the system and substrate stresses anddeformations vanish. This is not true however when a non-uniformtemperature is prescribed. When the limit is considered in this case thebare substrate still involves non-zero stresses and deformations. As aresult of this there seems to be “additional” interactions and couplingbetween the film and the substrate which are only active when anon-uniform temperature film is in existence.

The practical implications of the above are as follows: duringprocesses, such as various anneal or cooling steps when the temperaturevaries with time and across a film/substrate system, the former analysisis appropriate and should be used for the in-situ, real-time monitoringof film stress, through full-field curvature measurement. However, afterthe end of a process when the temperature field has equilibrated to auniform state, the latter analysis is of relevance and can be used toprovide the measurement of permanent (residual) stresses which have beenlocked in the film through the process and its non-uniformities (e.g.,residual effects of previously performed thermal treatment due to anon-uniform temperature distribution at the time of the treatment,effects of uneven doping in the film, and others). The latter analysisis also relevant for the study of non-uniform stress build up or relievein cases where temperature is not involved. These include certain typesof film deposition, etching or polishing process all of which can bemonitored by means of on-line full field curvature measurement methods.

The following sections first describe the analyzing methods for filmsystems with an axisymmetric ε_(m)(r) non-uniformity and then theanalyzing methods for film systems with an arbitrary ε_(m)(r,θ)non-uniformity and other non-uniformities. FIG. 1D shows the thin filmsystem model and the cylindrical coordinates used for the analysis.

In the following five titled sections, the equation numbering in eachsection is self contained and is independent from elsewhere in thisapplication.

1. Non-Uniform, Axisymmetric Misfit Strain in Thin Films Bonded on PlateSubstrates/Substrate Systems: The Relation between Non-Uniform FilmStresses and System Curvatures

Radially symmetric or axisymmetric variations in particular are oftenpresent in film/substrate systems. This is in part due to the circularwafer geometry and in part due to the axisymmetric geometries of mostprocessing equipment used to manufacture such wafers. An example ofaxisymmetric radial curvature distribution is given in FIG. 1E. FIGS.1E(a) and 1E(b) show the maximum and the minimum curvature distributions(principal curvature maps) of a large, industrial 300 mm wafer composedof a 1 μm thick low-k dielectric film deposited on a 730 μm Sisubstrate. This wafer was subjected to thermal anneal and the curvaturesshown correspond to curvature changes (after-before) following coolingfrom 400° C. The principal curvatures have been obtained by means of CGSinterferometry. Following optical measurement of the Cartesiancomponents, the principal curvatures K₁ and K₂ were obtained by using:$\begin{matrix}{\kappa_{1,2} = {\frac{\kappa_{xx} + \kappa_{yy}}{2} \pm {\left\{ {\left( \frac{\kappa_{xx} - \kappa_{yy}}{2} \right)^{2} + \kappa_{xy}^{2}} \right\}^{1/2}.}}} & (1.2)\end{matrix}$The wafer shape was not a priori assumed to be radially symmetric.However, the resulting principal curvature maps clearly show that thiswould be an accurate approximation in this case. The axisymmetry ofthese maps as well as the clear presence of large scale curvaturenon-uniformities, along the radial direction, provides strong motivationfor this study. This non-uniformity is in clear violation of Stoney's6^(th) assumption. Furthermore the two maps in FIG. 1E are clearlydifferent. This is also in clear violation of Stoney's 5^(th) assumptionwhich requires equi-biaxiality of curvature. To clarify the laststatement one should recall that once radial symmetry is established theonly two surviving components of curvature are${{\kappa_{r}(r)} = {{\frac{\mathbb{d}^{2}{w(r)}}{\mathbb{d}r^{2}}\quad{and}\quad{\kappa_{\theta}(r)}} = {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}}},$where z=w(r) is the equation of the radial wafer shape. With respect tothe polar system of FIG. 1D, κ_(r) and κ_(θ) are the radial andcircumferential curvature components respectively and are also equal tothe maximum and minimum principal curvatures. The remaining independentcurvature component (twist) vanishes along radial lines. Indeed in thiscase κ_(r)(r)≠κ_(θ)(r)∀R>r>0, clearly indicating that Stoney'sassumption of equi-biaxiality is violated.

This section considers the case of a thin film/substrate systemsubjected to arbitrary, radially symmetric misfit strain fields ε_(m)(r)in the thin film whose presence will create a radially symmetric stressand curvature field as well as arbitrarily large stress and curvaturegradients. Here the misfit strain refers to the intrinsic strain in thinfilm that is not associated with the stress. Our goal is to relate filmstresses and system curvatures to the misfit strain distribution and toultimately derive a relation between the film stresses and the systemcurvatures that would allow for the accurate experimental inference offilm stress from full-field and real-time curvature measurements.

A thin film deposited on a substrate is subject to axisymmetric misfitstrain distribution ε_(m)(r), where r is the radial coordinate (FIG.1D). The thin film and substrate are circular in the lateral directionand have a radius R.

The thin-film thickness h_(f) is much less than the substrate thicknessh_(s), and both are much less than R, i.e. h_(f)<<h_(s)<<R. The Young'smodulus and Poisson's ratio of the film and substrate are denoted byE_(f), v_(f), E_(s), and v_(s), respectively. The deformation isaxisymmetric and is therefore independent of the polar angle θ, where(r,θ,z) are cylindrical coordinates in the radial, circumferential (orazimuth) and the z directions with the origin at the center of thesubstrate (FIG. 1D).

The substrate is modeled as a plate since it can be subjected tobending, and h_(s)<<R. The thin film is modeled as a membrane whichcannot be subject to bending due to its small thickness h_(f)<<h_(s).Let u_(f)=u_(f)(r) denote the displacement in the radial (r) direction.The strains in the thin film are$ɛ_{rr} = {{\frac{\mathbb{d}u_{f}}{\mathbb{d}r}\quad{and}\quad ɛ_{\theta\theta}} = {\frac{u_{f}}{r}.}}$The stresses in the thin film can be obtained from the linear elasticconstitutive model as $\begin{matrix}{{\sigma_{rr} = {\frac{E_{f}}{1 - v_{f}^{2}}\left\lbrack {\frac{\mathbb{d}u_{f}}{\mathbb{d}r} + {v_{f}\frac{u_{f}}{r}} - {\left( {1 + v_{f}} \right)ɛ_{m}}} \right\rbrack}},{\sigma_{\theta\theta} = {{\frac{E_{f}}{1 - v_{f}^{2}}\left\lbrack {{v_{f}\frac{\mathbb{d}u_{f}}{\mathbb{d}r}} + \frac{u_{f}}{r} - {\left( {1 + v_{f}} \right)ɛ_{m}}} \right\rbrack}.}}} & (2.1)\end{matrix}$The membrane forces in the thin film areN_(r) ^((f))=h_(f)σ_(rr), N_(θ) ^((f))=h_(f)σ_(θθ).   (2.2)

It is recalled that, for uniform misfit strain ε_(m)(r)=constant, thenormal and shear stresses across the thin film/substrate interfacevanish except near the free edge r=R, i.e., σ_(zz)=σ_(rz)=0 at$z = {{\frac{h_{s}}{2}\quad{and}\quad r} < {R.}}$For non-uniform misfit strain ε_(m)=ε_(m)(r) as in the present study,the shear stress traction may not vanish anymore, and this shear stressσ_(rz) is denoted by τ(r) as shown in FIG. 1F. It is important to notethat the normal stress traction σ_(zz) still vanishes (except near thefree edge r=R) because the thin film cannot be subject to bending. Theequilibrium equation for the thin film, accounting for the effect ofinterface shear stress traction τ(r), becomes $\begin{matrix}{{\frac{\mathbb{d}\quad N_{r}^{(f)}}{\mathbb{d}r} + \frac{N_{r}^{(f)} - N_{\theta}^{(f)}}{r} - \tau} = 0.} & (2.3)\end{matrix}$The substitution of Eqs. (2.1) and (2.2) into (2.3) yields the followinggoverning equation for u_(f) (and τ) $\begin{matrix}{{\frac{\mathbb{d}^{2}u_{f}}{\mathbb{d}r^{2}} + {\frac{1}{r}\frac{\mathbb{d}u_{f}}{\mathbb{d}r}} - \frac{u_{f}}{r^{2}}} = {{\frac{1 - v_{f}^{2}}{E_{f}h_{f}}\tau} + {\left( {1 + v_{f}} \right){\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}.}}}} & (2.4)\end{matrix}$

Let u_(s) denote the displacement in the radial (r) direction at theneutral axis (z=0) of the substrate, and w the displacement in thenormal (z) direction. It is important to consider w since the substratecan be subject to bending and is modeled as a plate. The strains in thesubstrate are given by $\begin{matrix}{{ɛ_{rr} = {\frac{\mathbb{d}u_{s}}{\mathbb{d}r} - {z\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}}}},{ɛ_{\theta\theta} = {\frac{u_{s}}{r} - {z\quad\frac{1}{r}{\frac{\mathbb{d}w}{\mathbb{d}r}.}}}}} & (2.5)\end{matrix}$

The stresses in the substrate can then be obtained from the linearelastic constitutive model as $\begin{matrix}{{\sigma_{rr} = {\frac{E_{s}}{1 - v_{s}^{2}}\left\lbrack {\frac{\mathbb{d}u_{s}}{\mathbb{d}r} + {v_{s}\quad\frac{u_{s}}{r}} - {z\left( {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} + {\frac{v_{s}}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right)}} \right\rbrack}},{\sigma_{\theta\theta} = {{\frac{E_{s}}{1 - v_{s}^{2}}\left\lbrack {{v_{s}\frac{\mathbb{d}u_{s}}{\mathbb{d}r}} + \frac{u_{s}}{r} - {z\left( {{v_{s}\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} + {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right)}} \right\rbrack}.}}} & (2.6)\end{matrix}$

The forces and bending moments in the substrate are $\begin{matrix}{{N_{r}^{(s)} = {{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{\sigma_{rr}{\mathbb{d}z}}} = {\frac{E_{s}h_{s}}{1 - v_{s}^{2}}\left\lbrack {\frac{\mathbb{d}u_{s}}{\mathbb{d}r} + {v_{s}\frac{u_{s}}{r}}} \right\rbrack}}},{N_{\theta}^{(s)} = {{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{\sigma_{\theta\theta}{\mathbb{d}z}}} = {\frac{E_{s}h_{s}}{1 - v_{s}^{2}}\left\lbrack {{v_{s}\frac{\mathbb{d}u_{s}}{\mathbb{d}r}} + \frac{u_{s}}{r}} \right\rbrack}}},} & (2.7) \\{{M_{r} = {{- {\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{z\quad\sigma_{rr}{\mathbb{d}z}}}} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\left( {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} + {\frac{v_{s}}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right)}}},{M_{\theta} = {{- {\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{z\quad\sigma_{\theta\theta}{\mathbb{d}z}}}} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}{\left( {{v_{s}\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} + {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right).}}}}} & (2.8)\end{matrix}$

The shear stress τ at the thin film/substrate interface is equivalent tothe distributed axial force τ(r) and bending moment$\frac{h_{s}}{2}{\tau(r)}$applied at th e neutral axis (z=0) of the substrate. The in-plane forceequilibrium equation of the substrate then becomes $\begin{matrix}{{\frac{\mathbb{d}N_{r}^{(s)}}{\mathbb{d}r} + \frac{N_{r}^{(s)} - N_{\theta}^{(s)}}{r} + \tau} = 0.} & (2.9)\end{matrix}$The out-of-plane force and moment equilibrium equations are given by$\begin{matrix}{{{\frac{\mathbb{d}M_{r}}{\mathbb{d}r} + \frac{M_{r} - M_{\theta}}{r} + Q - {\frac{h_{s}}{2}\tau}} = 0},} & (2.10) \\{{{\frac{\mathbb{d}Q}{\mathbb{d}r} + \frac{Q}{r}} = 0},} & (2.11)\end{matrix}$where Q is the shear force normal to the neutral axis. The substitutionof Eq. (2.7) into Eq. (2.9) yields the following governing equation foru_(s) (and τ) $\begin{matrix}{{\frac{\mathbb{d}^{2}u_{s}}{\mathbb{d}r^{2}} + {\frac{1}{r}\frac{\mathbb{d}u_{s}}{\mathbb{d}r}} - \frac{u_{s}}{r^{2}}} = {{- \frac{1 - v_{s}^{2}}{E_{s}h_{s}}}{\tau.}}} & (2.12)\end{matrix}$Equation (2.11), together with the requirement of finite Q at the centerr=0, gives Q=0 in the entire substrate. Its substitution into Eq.(2.10), in conjunction with Eq. (2.8), gives the following governingequation for w (and τ) $\begin{matrix}{{\frac{\mathbb{d}^{3}w}{\mathbb{d}r^{3}} + {\frac{1}{r}\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} - {\frac{1}{r^{2}}\frac{\mathbb{d}w}{\mathbb{d}r}}} = {\frac{6\left( {1 - v_{s}^{2}} \right)}{E_{s}h_{s}^{2}}{\tau.}}} & (2.13)\end{matrix}$

The continuity of displacement across the thin film/substrate interfacerequires $\begin{matrix}{u_{f} = {u_{s} - {\frac{h_{s}}{2}{\frac{\mathbb{d}w}{\mathbb{d}r}.}}}} & (2.14)\end{matrix}$Equations (2.4) and (2.12)-(2.14) constitute four ordinary differentialequations for u_(f), u_(s), w and τ.

We can eliminate u_(f), u_(s) and w from these four equations to obtainthe shear stress at the thin film/substrate interface in terms of themisfit strain as $\begin{matrix}{{\tau = {{- \frac{\left( {1 + v_{f}} \right)}{\frac{1 - v_{\quad f}^{\quad 2}}{E_{f}h_{f}} + {4\frac{1 - v_{\quad s}^{\quad 2}}{E_{s}h_{s}}}}}\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}}},} & (2.15)\end{matrix}$which is a remarkable result that holds regardless of boundaryconditions at the edge r=R. Therefore, the interface shear stress isproportional to the gradient of misfit strain. For uniform misfit strainε_(m)(r)=constant, the interface shear stress vanishes, i.e., τ=0.

The substitution of the above solution for shear stress τ into Eqs.(2.13) and (2.12) yields ordinary differential equations fordisplacements w and u_(s) in the substrate. Their general solutions are$\begin{matrix}{{\frac{\mathbb{d}w}{\mathbb{d}r} = {{{- \frac{6\left( {1 - v_{s}^{2}} \right)}{E_{s}h_{s}^{2}}}\frac{1 + v_{f}}{\frac{1 - v_{\quad f}^{\quad 2}}{E_{f}h_{f}} + {4\frac{1 - v_{\quad s}^{\quad 2}}{E_{s}h_{s}}}}\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{B_{1}}{2}r}}},} & (2.16) \\{{u_{s} = {{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{1 + v_{f}}{\frac{1 - v_{\quad f}^{\quad 2}}{E_{f}h_{f}} + {4\frac{1 - v_{\quad s}^{\quad 2}}{E_{s}h_{s}}}}\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{B_{2}}{2}r}}},} & (2.17)\end{matrix}$where B₁ and B₂ are constants to be determined by boundary conditions tobe given in the next section. We have imposed the conditions that w andu_(s) are bounded at the center of the substrate r=0. The displacementu_(f) in the thin film can be obtained from interface continuitycondition in Eq. (2.14) as $\begin{matrix}{u_{f} = {{\frac{4\left( {1 - v_{s}^{2}} \right)}{E_{s}h_{s}}\frac{\left( {1 + v_{f}^{2}} \right)}{\frac{1 + v_{f}^{2}}{E_{f}h_{f}} + {4\frac{41 - v_{s}^{2}}{E_{s}h_{s}}}}\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\left( {\frac{\quad B_{\quad 2}}{\quad 2} - \frac{\quad{h_{\quad s}\quad B_{\quad 1}}}{\quad 4}} \right){r.}}}} & (2.18)\end{matrix}$

The force N_(r) ^((f)) in the thin film, which is needed for boundaryconditions in the next section, is obtained from Eq. (2.2) as$\begin{matrix}{N_{r}^{(f)} = {\frac{E_{f}h_{f}}{1 - v_{f}^{2}}{\begin{Bmatrix}\begin{matrix}{{{- \left( {1 + v_{f}} \right)}\frac{\frac{1 - v_{f}^{2}}{E_{f}h_{f}}ɛ_{m}}{\frac{1 - v_{f}^{2}}{E_{f}h_{f}} + {4\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}}} -} \\{{4\left( {1 - v_{f}^{2}} \right)\frac{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}ɛ_{m}}{\frac{1 - v_{f}^{2}}{E_{f}h_{f}} + {4\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}}\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +}\end{matrix} \\{\frac{1 + v_{f}}{2}\left( {B_{2} - {\frac{h_{s}}{2}B_{1}}} \right)}\end{Bmatrix}.}}} & (2.19)\end{matrix}$The force N_(r) ^((s)) and moment M_(r) in the substrate, which are alsoneeded for boundary conditions in the next section, are obtained fromEqs. (2.7) and (2.8) as $\begin{matrix}{{N_{r}^{(s)} = {\frac{E_{s}h_{s}}{1 - v_{s}^{2}}\begin{Bmatrix}\begin{matrix}{{\left( {1 + v_{f}} \right)\frac{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}ɛ_{m}}{\frac{1 - v_{f}^{2}}{E_{f}h_{f}} + {4\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}}} -} \\{{\left( {1 - v_{s}} \right)\left( {1 + v_{f}} \right)\frac{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}{\frac{1 - v_{f}^{2}}{E_{f}h_{f}} + {4\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}}\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +}\end{matrix} \\{\frac{1 + v_{s}}{2}B_{2}}\end{Bmatrix}}},} & (2.20) \\{M_{r} = {\frac{h_{s}}{2}{\begin{Bmatrix}\begin{matrix}{{{- \left( {1 + v_{f}} \right)}\frac{ɛ_{m}}{\frac{1 - v_{f}^{2}}{E_{f}h_{f}} + {4\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}}} +} \\{{\left( {1 - v_{s}} \right)\left( {1 + v_{f}} \right)\frac{1}{\frac{1 - v_{f}^{2}}{E_{f}h_{f}} + {4\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}}\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +}\end{matrix} \\{\frac{E_{s}h_{s}^{2}}{12\left( {1 - v_{s}} \right)}B_{1}}\end{Bmatrix}.}}} & (2.21)\end{matrix}$

It is interesting to observe that, in the limit of h_(f)/h_(s)<<1, thedisplacements in Eqs. (2.16)-(2.18) become $\begin{matrix}{{\frac{\mathbb{d}w}{\mathbb{d}r} = {{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{B_{1}}{2}r} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},} & (2.22) \\{{u_{s} = {{\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{B_{2}}{2}r} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},} & (2.23) \\{u_{f} = {{4\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\begin{pmatrix}{\frac{B_{2}}{2} -} \\\frac{h_{s}B_{1}}{4}\end{pmatrix}r} + {{O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}.}}} & (2.24)\end{matrix}$

The boundary conditions for the analysis are as follows. The firstboundary condition at the free edge r=R requires that the net forcevanishes,N _(r) ^((f)) +N _(r) ^((s))=0 at r=R.   (3.1)The above equation, in conjunction with Eqs. (2.19) and (2.20), gives$\begin{matrix}{\frac{B_{2}}{2} = {{\frac{E_{f}h_{f}}{1 - v_{f}}\frac{\left( {1 - v_{s}} \right)^{2}}{E_{s}h_{s}}\frac{1}{R^{2}}{\int_{0}^{R}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}} & (3.2)\end{matrix}$under the limit ε=h_(f)/h_(s)<<1. The second boundary condition at thefree edge r=R is vanishing of net moment, i.e., $\begin{matrix}{{M_{r} - {\frac{h_{s}}{2}N_{r}^{(f)}}} = {{0\quad{at}\quad r} = {R.}}} & (3.3)\end{matrix}$In conjunction with Eqs. (2.22), (2.24) and (3.2), the above equationgives $\begin{matrix}{\frac{B_{1}}{2} = {{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{\left( {1 - v_{s}} \right)^{2}}{E_{s}h_{s}^{2}}\frac{1}{R^{2}}{\int_{0}^{R}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {{O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}.}}} & (3.4)\end{matrix}$

It is important to point out that the boundary conditions can also beestablished from the variational principle (e.g., Freund, 2000). Thetotal potential energy in the thin film/substrate system with the freeedge at r=R is $\begin{matrix}{{\Pi = {2\pi{\int_{0}^{R}{r\quad{\mathbb{d}r}{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2} + h_{f}}{U\quad{\mathbb{d}z}}}}}}},} & (3.5)\end{matrix}$where U is the strain energy density which gives$\frac{\partial U}{\partial ɛ_{rr}} = {{\sigma_{rr}\quad{and}\quad\frac{\partial U}{\partial ɛ_{\theta\theta}}} = {\sigma_{\theta\theta}.}}$For constitutive relations in Eqs. (2.1) and (2.6), we obtain$\begin{matrix}{{U = {\frac{E}{2\left( {1 - v^{2}} \right)}\left\lbrack {ɛ_{rr}^{2} + ɛ_{\theta\theta}^{2} + {2v\quad ɛ_{rr}ɛ_{\theta\theta}} - {2\left( {1 + v} \right){ɛ_{m}\left( {ɛ_{rr} + ɛ_{\theta\theta}} \right)}}} \right\rbrack}},} & (3.6)\end{matrix}$where E and v take their corresponding values in the thin film (i.e.,E_(f) and v_(f) for$\left. {{\frac{h_{s}\bullet}{2} + h_{f}} \geq z \geq \frac{h_{s}\bullet}{2}} \right)$and in the substrate (i.e., E_(s) and v_(s) for$\left. {\frac{h_{s}\bullet}{2} \geq z \geq {- \frac{h_{s}\bullet}{2}}} \right),$and ε_(m) is zero in the substrate. For the displacement field inSection2 and the associated strain field, the potential energy Π in Eq.(3.5) becomes a quadratic function of parameters B₁ and B₂. Theprinciple of minimal potential energy requires $\begin{matrix}{\frac{\partial\Pi}{\partial B_{1}} = {{0\quad{and}\quad\frac{\partial\Pi}{\partial B_{2}}} = 0.}} & (3.7)\end{matrix}$It can be shown that, as expected in the limit h_(f)/h_(s)<<1, the abovetwo equations are equivalent to the vanishing of net force in Eq. (3.1)and net moment in Eq. (3.3). The displacements in Eqs. (2.22)-(2.24) nowbecome $\begin{matrix}{{\frac{\mathbb{d}w}{\mathbb{d}r} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\frac{1}{r}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{r}{R^{2}}{\int_{0}^{R}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}}},} & (3.8) \\{\begin{matrix}{u_{f} = {4\quad u_{s}}} \\{= {4\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\left\lbrack {{\frac{1}{r}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{r}{R^{2}}{\int_{0}^{R}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}}}\end{matrix}{{{for}\quad{h_{f}/h_{s}}} ⪡ 1.}} & (3.9)\end{matrix}$

We now examine the stresses and curvatures in thin film and substrate.The substrate curvatures can be obtained from the displacement w as$\begin{matrix}{\begin{matrix}{\kappa_{r} = \frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} \\{{= {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {ɛ_{m} - {\frac{1}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{1}{R^{2}}{\int_{0}^{R}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}}},}\end{matrix}\begin{matrix}{\kappa_{\theta} = {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \\{= {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\frac{1}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{1}{R^{2}}{\int_{0}^{R}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}.}}}\end{matrix}} & (4.1)\end{matrix}$The sum of these two curvatures is $\begin{matrix}{{{\kappa_{r} + \kappa_{\theta}} = {{- 12}\quad\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}}{E_{s}h_{s}^{2}}\left\lbrack {\overset{\_}{ɛ_{m}} + {\frac{1 + v_{s}}{2}\left( {ɛ_{m} - \overset{\_}{ɛ_{m}}} \right)}} \right\rbrack}}},{{or}\quad{equivalently}}} & (4.2) \\{{{\kappa_{r} + \kappa_{\theta}} = {{- 12}\quad\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}}{E_{s}h_{s}^{2}}\left\lbrack {ɛ_{m} - {\frac{1 - v_{s}}{2}\left( {ɛ_{m} - \overset{\_}{ɛ_{m}}} \right)}} \right\rbrack}}},} & (4.3)\end{matrix}$where$\overset{\_}{ɛ_{m}} = {{\frac{2}{R^{2}}{\int_{0}^{R}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} = \frac{\int{\int{ɛ_{m}{\mathbb{d}A}}}}{\pi\quad R^{2}}}$is the average misfit strain in the thin film. The first term on theright hand side of Eq. (4.2) corresponds to a uniform (average) misfitstrain, while the second term gives the deviation from the uniformmisfit strain. Such a deviation is proportional to the differencebetween the local misfit strain ε_(m) and the average misfit strainε_(m) . Similarly, the first term on the right hand side of Eq. (4.3)corresponds to the local misfit strain ε_(m), while the second termgives the deviation from the local misfit strain and is alsoproportional to ε_(m)− ε_(m) .

The difference between two curvatures in Eq. (4.1) is $\begin{matrix}{{\kappa_{r} - \kappa_{\theta}} = {{- 6}\quad\frac{E_{f}h_{f}}{1 - v_{f}}{{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {ɛ_{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}.}}} & (4.4)\end{matrix}$The forces in the substrate are obtained from Eq. (2.7) as$\begin{matrix}{{N_{r}^{(s)} = {\frac{E_{f}h_{f}}{1 - v_{f}}\left\{ {ɛ_{m} - {\frac{1 - v_{s}}{2}\left\lbrack {{\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} - \overset{\_}{ɛ_{m}}} \right\rbrack}} \right\}}},{N_{\theta}^{(s)} = {\frac{E_{f}h_{f}}{1 - v_{f}}\left\{ {{v_{s}ɛ_{m}} + {\frac{1 - v_{s}}{2}\left\lbrack {{\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + \overset{\_}{ɛ_{m}}} \right\rbrack}} \right\}}}} & (4.5)\end{matrix}$for h_(f)/h_(s)<<1. The bending moments in the substrate are obtainedfrom Eq. (2.8) as $\begin{matrix}{{M_{r} = {\frac{E_{f}h_{f}}{1 - v_{f}}\frac{h_{s}}{2}\left\{ {{- ɛ_{m}} + {\frac{1 - v_{s}}{2}\left\lbrack {{\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} - \overset{\_}{ɛ_{m}}} \right\rbrack}} \right\}}},{M_{\theta} = {\frac{E_{f}h_{f}}{1 - v_{f}}\frac{h_{s}}{2}\left\{ {{{- v_{s}}ɛ_{m}} - {\frac{1 - v_{s}}{2}\left\lbrack {{\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + \overset{\_}{ɛ_{m}}} \right\rbrack}} \right\}}}} & (4.6)\end{matrix}$for h_(f)/h_(s)<<1. The stresses in the substrate are related to theforces and moments by $\begin{matrix}{{\sigma_{r\quad r}^{(s)} = {\frac{N_{r}^{(s)}}{h_{s}} - {\frac{12\quad M_{r}}{h_{s}^{3}}z}}},{\sigma_{\theta\quad\theta}^{(s)} = {\frac{N_{\theta}^{(s)}}{h_{s}} - {\frac{12\quad M_{\theta}}{h_{s}^{3}}{z.}}}}} & (4.7)\end{matrix}$

The stresses in the thin film are obtained from Eq. (2.1),$\begin{matrix}{{\sigma_{r\quad r}^{(f)} = {{\frac{E_{f}}{1 - v_{f}}\begin{Bmatrix}{{- ɛ_{m}} + {4\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}} \\\begin{bmatrix}{ɛ_{m} - {\left( {1 - v_{f}} \right)\frac{1}{\quad r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} +} \\{\frac{1 + v_{f}}{\quad 2}\frac{1 - v_{s}}{\quad{1 + v_{s}}}\quad\overset{\_}{ɛ_{m}}}\end{bmatrix}\end{Bmatrix}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},{\sigma_{\theta\quad\theta}^{(f)} = {{\frac{E_{f}}{1 - v_{f}}\begin{Bmatrix}{{- ɛ_{m}} + {4\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}}} \\\begin{bmatrix}{{v_{f}ɛ_{m}} + {\left( {1 - v_{f}} \right)\frac{1}{\quad r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} +} \\{\frac{1 + v_{f}}{\quad 2}\frac{1 - v_{s}}{\quad{1 + v_{s}}}\quad\overset{\_}{ɛ_{m}}}\end{bmatrix}\end{Bmatrix}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},} & (4.8)\end{matrix}$where we have kept the terms that are up to the order of$O\left( \frac{h_{f}}{h_{s}} \right)$in order to illustrate the difference between these two stresses. Thesum and difference of these stresses have the following simpleexpressions $\begin{matrix}{\quad{{{\sigma_{r\quad r}^{(f)} + \sigma_{\theta\quad\theta}^{(f)}} = {{\frac{E_{f}}{1 - v_{f}}\left( {{- 2}\quad ɛ_{m}} \right)} + {O\left( \frac{h_{f}}{h_{s}} \right)}}},{{\sigma_{r\quad r}^{(f)} + \sigma_{\theta\quad\theta}^{(f)}} = {{4\quad E_{f}\frac{E_{f}h_{f}}{1 - v_{f}^{2}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\left\lbrack {ɛ_{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}} + {{O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}.}}}}} & (4.9)\end{matrix}$It is noted that σ_(rr) ^((f))−σ_(θθ) ^((f)) is in general expected tobe smaller than σ_(rr) ^((f))+σ_(θθ) ^((f)) for h_(f)/h_(s)<<1. Thisissue will be discussed again 10 in the next section in relation to thedependence of this stress difference to the difference between polarcurvature components.

For uniform misfit strain ε_(m)=constant, the curvatures in thesubstrate obtained from Eqs. (4.2)-(4.4) become$\kappa = {\kappa_{r} = {\kappa_{\theta} = {{- 6}\quad\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}{ɛ_{m}.}}}}$The stresses in the thin film obtained from Eq. (4.9) become$\sigma = {\sigma_{r\quad r}^{(f)} = {\sigma_{\theta\quad\theta}^{(f)} = {\frac{E_{f}}{1 - v_{f}}{\left( {- ɛ_{m}} \right).}}}}$For this special case only, both stress and curvature states becomeequi-biaxial. The elimination of misfit strain ε_(m) from the above twoequations yields a simple relation$\sigma = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}{\kappa.}}$This is the relation obtained by Stoney [see Eq. (1.1)] and it has beenused to estimate the thin-film stress a from the substrate curvature κ,if the misfit strain, stress and curvature are all constants, and if theplate system shape is spherical. In the following, we extend such arelation for non-uniform misfit strain distribution.

We now describe the extension of Stoney formula for non-uniform misfitstrain distribution. The stresses and curvatures are all given in termsof the misfit strain in the previous section. We extend the Stoneyformula for non-uniform misfit distribution in this section byestablishing the direct relation between the thin-film stresses andsubstrate curvatures.

It is shown that both κ_(r)−κ_(θ) in Eq. (4.4) and σ_(rr) ^((f))−σ_(θθ)^((f)) in Eq. (4.9) are proportional to${ɛ_{m}(r)} - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{{\mathbb{d}\eta}.}}}}$Therefore, elimination of misfit strain gives the difference σ_(rr)^((f))−σ_(θθ) ^((f)) in thin-film stresses directly proportional to thedifference κ_(r)−κ_(θ) in substrate curvatures, $\begin{matrix}{\sigma_{r\quad r}^{(f)} = {\sigma_{\theta\quad\theta}^{(f)} = {{- \frac{2\quad E_{f}h_{s}}{3\left( {1 + v_{f}} \right)}}{\left( {\kappa_{r} - \kappa_{\theta}} \right).}}}} & (5.1)\end{matrix}$

The above relation clearly shows that the radial and circumferentialstress components will be equal only if the equivalent curvaturecomponents are also equal (Unlike what was stated in Stoney's 5^(th) and6^(th) assumptions this is not true in general as can be clearly seen byFIG. 2. In this experimental example, the two polar components ofcurvature displayed are radially varying but are clearly unequal.

We now focus on the sum of thin-film stresses σ_(rr) ^((f))+σ_(θθ)^((f)) and sum of substrate curvatures κ_(r)+κ_(θ). We define theaverage substrate curvature κ_(r)+κ_(θ) as $\begin{matrix}{\overset{\_}{\kappa_{r} + \kappa_{\theta}} = {{\frac{1}{\pi\quad R^{2}}\underset{A}{\int\int}\left( {\kappa_{r} + \kappa_{\theta}} \right)\eta{\mathbb{d}\eta}{\mathbb{d}\theta}} = {\frac{2}{R^{2}}{\int_{0}^{R}{{\eta\left( {\kappa_{r} + \kappa_{\theta}} \right)}{{\mathbb{d}\eta}.}}}}}} & (5.2)\end{matrix}$It can be related to the average misfit strain ε_(m) by averaging bothsides of Eq. (4.3), i.e., $\begin{matrix}{\overset{\_}{\kappa_{r} + \kappa_{\theta}} = {12\quad\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}{\left( {- \overset{\_}{ɛ_{m}}} \right).}}} & (5.3)\end{matrix}$The deviation from the average curvature, κ_(r)+κ_(θ)− κ_(r)+κ_(θ) , canbe related to the deviation from the average misfit strain ε_(m)− ε_(m)as $\begin{matrix}{{\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}} = {{- 6}\quad\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}{\left( {ɛ_{m} - \overset{\_}{ɛ_{m}}} \right).}}} & (5.4)\end{matrix}$The elimination of misfit strain ε_(m)− ε_(m) and average misfit strainε_(m) from Eqs. (5.3), (5.4) and (4.9) gives the sum of thin-filmstresses in terms of curvature as $\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}{\left\{ {\kappa_{rr} + \kappa_{\theta\theta} + {\frac{1 - v_{s}}{1 + v_{s}}\left\lbrack {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}} \right\rbrack}} \right\}.}}} & (5.5)\end{matrix}$The above equation, together with Eq. (5.1), provide direct relationsbetween thin-film stresses and substrate curvatures. It is important tonote that stresses at a point in the thin film depend not only oncurvatures at the same point (local dependence), but also on the averagecurvature in the entire substrate (non-local dependence).

The interface stress τ(r) gives in Eq. (2.15) can also be directlyrelated to substrate curvatures via $\begin{matrix}{\tau = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\frac{\mathbb{d}}{\mathbb{d}r}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right).}}} & (5.6)\end{matrix}$This provides a remarkably simple way to estimate the interface shearstress from radial gradients of the two non-zero substrate curvatures.

Since interfacial shear stresses are responsible for promoting systemfailures through delamination of the thin film from the substrate, Eq.(5.6) has particular significance. It shows that such stresses areproportional to the radial gradient of κ_(rr)+κ_(θθ) and not to itsmagnitude as might have been expected of a local, Stoney-likeformulation. The implementation value of Eq. (5.6) is that it providesan easy way of inferring these special interfacial shear stresses oncethe full-field curvature information is available. As a result, themethodology also provides a way to evaluate the risk of and to mitigatesuch important forms of failure. It should be noted that for the specialcase of spatially constant curvatures, this interfacial shear stress Tvanishes as is the case for all Stoney-like formulations described inthe introduction.

In addition, Eq. (5.5) also reduces to Stoney's result for the case ofspatial curvature uniformity. Indeed for this case, Eq. (5.5) reducesto: $\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right).}}} & (5.7)\end{matrix}$If in addition the curvature state is equi-biaxial (κ_(rr)=κ_(θθ)), asassumed by Stoney, the classic Stoney equation emerges from Eq. (5.7)while relation (5.1) furnishes σ_(rr)=σ_(θθ) (stress equi-biaxiality) asa special case.

Unlike Stoney's original analysis technique and some prior extensions ofthe Stoney's analysis technique, the present analysis shows that thedependence of the stresses on the curvatures is not generally “local.”Here the stress components at a point on the film will, in general,depend on both the local value of the curvature components (at the samepoint) and on the value of curvatures of all other points on the platesystem (non-local dependence). The more pronounced the curvaturenon-uniformities are, the more important such non-local effects becomein accurately determining film stresses from curvature measurements.This demonstrates that analyses methods based on Stoney's approach andits various extensions cannot handle the non-locality of thestress/curvature dependence and may result in substantial stressprediction errors if such analyses are applied locally in cases wherespatial variations of system curvatures and stresses are present.

The presence of non-local contributions in such relations also hasimplications regarding the nature of diagnostic methods needed toperform wafer-level film stress measurements. Notably the existence ofnon-local terms necessitates the use of full-field methods capable ofmeasuring curvature components over the entire surface of the platesystem (or wafer). Furthermore measurement of all independent componentsof the curvature field is necessary. This is because the stress state ata point depends on curvature contributions (from both κ_(rr) and κ_(θθ))from the entire plate surface.

Regarding the curvature-misfit strain [Eqs. (4.1)-(4.4)] andstress-misfit strain [Eqs. (4.8) -(4.10)] relations the following pointsare noteworthy. These relations also generally feature a dependence oflocal misfit strain ε_(m)(r) which is “Stoney-like” as well as a“non-local” contribution from the misfit strain of other points on theplate system. Furthermore the stress and curvature states are alwaysnon-equibiaxial (i.e., σ_(rr) ^((f))≠σ_(θθ) ^((f)) and K_(rr)≠K_(θθ)) inthe presence of misfit strain non-uniformities. Only if ε_(m)=constantthese states become equi-biaxial, the “non-local” contributions vanishand Stoney's original results are recovered as a special case and ahighly unlikely scenario as clearly demonstrated from FIG. 1E.

The existence of radial non-uniformities also results in theestablishment of shear stresses along the film/substrate interface.These stresses are in general proportional to the radial derivatives ofthe first curvature invariant κ_(rr)+κ_(θθ) [Eq. (5.6)]. In terms ofmisfit strain these interfacial shear stresses are also proportional tothe radial gradient of the misfit strain distribution ε_(m)(r). Theoccurrence of such stresses is ultimately related to spatialnon-uniformities and as a result such stresses vanish for the specialcase of uniform κ_(rr)+κ_(θθ) or ε_(m) considered by Stoney and itsvarious extensions. Since film delamination is a commonly encounteredform of failure during wafer manufacturing, the ability to estimate thelevel and distribution of such stresses from wafer-level metrology mightprove to be invaluable in enhancing the reliability of such systems.

2. Spatially Non-Uniform, Isotropic Misfit Strain in Thin Films Bondedon Plate Substrates: The Relation Between Non-Uniform Film Stresses andSystem Curvatures

We now describe analysis of a thin film deposited on a substrate that issubject to arbitrary misfit strain distribution ε^(m)(r,θ), where r andθ are the polar coordinates (FIG. 1D). The thin film and substrate arecircular in the lateral direction and have a radius R. Like in theprevious section, the thin-film thickness h_(f) is much less than thesubstrate thickness h_(s), and both are much less than R, i.e.h_(f)<<h_(s)<<R. The Young's modulus and Poisson's ratio of the film andsubstrate are denoted by E_(f),v_(f),E_(s) and v_(s), respectively. Thesubstrate is modeled as a plate since it can be subjected to bending,and h_(s)<<R. The thin film is modeled as a membrane which cannot besubject to bending due to its small thickness h_(f)<<h_(s).

Let u_(r) ^((f)) and u_(θ) ^((f)) denote the displacements in the radial(r) and circumferential (θ) directions. The strains in the thin film are${ɛ_{rr} = \frac{\partial u_{r}^{(f)}}{\partial r}},{ɛ_{\theta\theta} = {\frac{u_{r}^{(f)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(f)}}{\partial\theta}}}},$and$\gamma_{r\quad\theta} = {{\frac{1}{r}\frac{\partial u_{r}^{(f)}}{\partial\theta}} + \frac{\partial u_{\theta}^{(f)}}{\partial r} - {\frac{u_{\theta}^{(f)}}{r}.}}$The strains in the film are related to the stresses and the misfitstrain ε^(m) by$ɛ_{ij} = {{\frac{1}{E_{f}}\left\lbrack {{\left( {1 + v_{f}} \right)\sigma_{ij}} - {v_{f}\sigma_{kk}\delta_{ij}}} \right\rbrack} + {ɛ^{m}\delta_{ij}}}$via the linear elastic constitutive model, which can be equivalentlywritten as $\begin{matrix}{{\sigma_{rr} = {\frac{E_{f}}{1 - v_{f}^{2}}\left\lbrack {\frac{\partial u_{r}^{(f)}}{\partial r} + {v_{f}\left( {\frac{u_{r}^{(f)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(f)}}{\partial\theta}}} \right)} - {\left( {1 + v_{f}} \right)ɛ^{m}}} \right\rbrack}},{\sigma_{\theta\theta} = {\frac{E_{f}}{1 - v_{f}^{2}}\left\lbrack {{v_{f}\frac{\partial u_{r}^{(f)}}{\partial r}} + \frac{u_{r}^{(f)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(f)}}{\partial\theta}} - {\left( {1 + v_{f}} \right)ɛ^{m}}} \right\rbrack}},{\sigma_{r\quad\theta} = {\frac{E_{f}}{2\left( {1 + v_{f}^{2}} \right)}{\left( {{\frac{1}{r}\frac{\partial u_{r}^{(f)}}{\partial\theta}} + \frac{\partial u_{\theta}^{(f)}}{\partial r} - \frac{u_{\theta}^{(f)}}{r}} \right).}}}} & (2.1)\end{matrix}$The membrane forces in the thin film areN _(r) ^((f)) =h _(f)σ_(rr) , N _(θ) ^((f)) =h _(f)σ_(θθ) , N _(rθ)^((f)) =h _(f)σ_(rθ).   (2.2)

For uniform misfit strain distribution ε^(m)=constant, the normal andshear stresses across the thin film/substrate interface vanish exceptnear the free edge r=R, i.e., σ_(zz)=σ_(rz)=σ_(rθ)=0 at$z = \frac{h_{s}}{2}$and r<R. For non-uniform misfit strain distribution ε^(m)=ε^(m)(r,θ),the shear stress σ_(rz) and σ_(θz) at the interface may not vanishanymore, and are denoted by τ_(r) and τ_(θ), respectively. It isimportant to note that the normal stress traction σ_(zz) still vanishes(except near the free edge r=R) because the thin film cannot be subjectto bending. The equilibrium equations for the thin film, accounting forthe effect of interface shear stresses τ_(r) and τ₀, become$\begin{matrix}{{{\frac{\partial N_{r}^{(f)}}{\partial r} + \frac{N_{r}^{(f)} - N_{\theta}^{(f)}}{r} + {\frac{1}{r}\frac{\partial N_{r\quad\theta}^{(f)}}{\partial\theta}} - \tau_{r}} = 0},{{\frac{\partial N_{r\quad\theta}^{(f)}}{\partial\theta} + {\frac{2}{r}N_{r\quad\theta}^{(f)}} + {\frac{1}{r}\frac{\partial N_{\theta}^{(f)}}{\partial\theta}} - \tau_{\theta}} = 0.}} & (2.3)\end{matrix}$The substitution of Eqs. (2.1) and (2.2) into (2.3) yields the followinggoverning equations for u_(r) ^((f)),u_(θ) ^((f)),τ_(r) and τ_(θ)$\begin{matrix}{{{\frac{\partial^{2}u_{r}^{(f)}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{r}^{(f)}}{\partial r}} - \frac{u_{r}^{(f)}}{r^{2}} + {\frac{1 - v_{f}}{2}\frac{1}{r^{2}}\frac{\partial^{2}u_{r}^{(f)}}{\partial\theta^{2}}} + {\frac{1 + v_{f}}{2}\frac{1}{r}\frac{\partial^{2}u_{\theta}^{(f)}}{{\partial r}{\partial\theta}}} - {\frac{3 - v_{f}}{2}\frac{1}{r^{2}}\frac{\partial u_{\theta}^{(f)}}{\partial\theta}}} = {{\frac{1 - v_{f}^{2}}{E_{f}h_{f}}\tau_{r}} + {\left( {1 + v_{f}} \right)\frac{\partial ɛ^{m}}{\partial r}}}},{{{\frac{1 + v_{f}}{2}\frac{1}{r}\frac{\partial^{2}u_{r}^{(f)}}{{\partial r}{\partial\theta}}} + {\frac{3 - v_{f}}{2}\frac{1}{r^{2}}\frac{\partial u_{r}^{(f)}}{\partial\theta}} + {\frac{1 - v_{f}}{2}\left( {\frac{\partial^{2}u_{\theta}^{(f)}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{\theta}^{(f)}}{\partial r}} - \frac{u_{\theta}^{(f)}}{r^{2}}} \right)} + {\frac{1}{r^{2}}\frac{\partial^{2}u_{\theta}^{(f)}}{\partial\theta^{2}}}} = {{\frac{1 - v_{f}^{2}}{E_{f}h_{f}}\tau_{\theta}} + {\left( {1 + v_{f}} \right)\frac{1}{r}{\frac{\partial ɛ^{m}}{\partial\theta}.}}}}} & (2.4)\end{matrix}$

Let u_(r) ^((s)) and u_(θ) ^((s)) denote the displacements in the radial(r) and circumferential (θ) directions at the neutral axis (z=0) of thesubstrate, and w the displacement in the normal (z) direction. It isimportant to consider w since the substrate can be subject to bendingand is modeled as a plate. The strains in the substrate are given by$\begin{matrix}{{ɛ_{rr} = {\frac{\partial u_{r}^{(s)}}{\partial r} - {z\frac{\partial^{2}w}{\partial r^{2}}}}},{ɛ_{\theta\theta} = {\frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}} - {z\left( {{\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}}},{\gamma_{r\quad\theta} = {{\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial\theta}} + \frac{\partial u_{\theta}^{(s)}}{\partial r} - \frac{u_{\theta}^{(s)}}{r} - {2z\frac{\partial}{\partial r}{\left( {\frac{1}{r}\frac{\partial w}{\partial\theta}} \right).}}}}} & (2.5)\end{matrix}$The stresses in the substrate can then be obtained from the linearelastic constitutive model as $\begin{matrix}{{\sigma_{rr} = {\frac{E_{s}}{1 - v_{s}^{2}}\begin{Bmatrix}{\frac{\partial u_{r}^{(s)}}{\partial r} + {v_{s}\left( {\frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right)} -} \\{z\left\lbrack {\frac{\partial^{2}w}{\partial r^{2}} + {v_{s}\left( {{\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}} \right\rbrack}\end{Bmatrix}}},{\sigma_{\theta\theta} = {\frac{E_{s}}{1 - v_{s}^{2}}\begin{bmatrix}{{v_{s}\frac{\partial u_{r}^{(s)}}{\partial r}} + \frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}} -} \\{z\left( {{v_{s}\frac{\partial^{2}w}{\partial r^{2}}} + {\frac{1}{r^{2}}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}\end{bmatrix}}},{\theta_{r\quad\theta} = {{\frac{E_{s}}{2\left( {1 + v_{s}} \right)}\left\lbrack {{\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial\theta}} + \frac{\partial u_{\theta}^{(s)}}{\partial r} - \frac{u_{\theta}^{(s)}}{r} - {2z\frac{\partial}{\partial r}\left( {\frac{1}{r}\frac{\partial w}{\partial\theta}} \right)}} \right\rbrack}.}}} & (2.6)\end{matrix}$The forces and bending moments in the substrate are $\begin{matrix}{{N_{r}^{(s)} = {{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{\sigma_{rr}\quad{\mathbb{d}z}}} = {\frac{E_{s}h_{s}}{1 - v_{s}^{2}}\left\lbrack {\frac{\partial u_{r}^{(s)}}{\partial r} + {v_{s}\left( {\frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right)}} \right\rbrack}}},{N_{\theta}^{(s)} = {{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{\sigma_{\theta\theta}\quad{\mathbb{d}z}}} = {\frac{E_{s}h_{s}}{1 - v_{s}^{2}}\left( {{v_{s}\frac{\partial u_{r}^{(s)}}{\partial r}} + \frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right)}}},{N_{r\quad\theta}^{(s)} = {{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{\sigma_{r\quad\theta}\quad{\mathbb{d}z}}} = {\frac{E_{s}h_{s}}{2\left( {1 + v_{s}} \right)}{\left( {{\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial\theta}} + \frac{\partial u_{\theta}^{(s)}}{\partial r} - \frac{u_{\theta}^{(s)}}{r}} \right).}}}}} & (2.7) \\{{{M_{r} = {{- {\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{z\quad\sigma_{rr}\quad{\mathbb{d}z}}}} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\begin{bmatrix}{\frac{\partial^{2}w}{\partial r^{2}} +} \\{v_{s}\left( {{\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}\end{bmatrix}}}},\begin{matrix}{M_{\theta} = {- {\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{z\quad\sigma_{\theta\theta}\quad{\mathbb{d}z}}}}} \\{{= {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\left( {{v_{s}\frac{\partial^{2}w}{\partial r^{2}}} + {\frac{1}{r^{2}}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}},}\end{matrix}}{M_{r\quad\theta} = {{- {\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2}}{z\quad\sigma_{r\quad\theta}\quad{\mathbb{d}z}}}} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 + v_{s}} \right)}\frac{\partial}{\partial r}{\left( {\frac{1}{r}\frac{\partial w}{\partial\theta}} \right).}}}}} & (2.8)\end{matrix}$

The shear stresses τ_(r) and τ_(θ) at the thin film/substrate interfaceare equivalent to the distributed forces τ_(r) in the radial directionand τ_(θ) in the circumferential direction, and bending moments$\frac{h_{s}}{2}\tau_{r}\quad{and}\quad\frac{h_{s}}{2}\tau_{\theta}$applied at the neutral axis (z=0) of the substrate. The in-plane forceequilibrium equations of the substrate then become $\begin{matrix}{{{\frac{\partial N_{r}^{(s)}}{\partial r} + \frac{N_{r}^{(s)} - N_{\theta}^{(s)}}{r} + {\frac{1}{r}\frac{\partial N_{r\quad\theta}^{(s)}}{\partial\theta}} + \tau_{r}} = 0},{{\frac{\partial N_{r\quad\theta}^{(s)}}{\partial r} + {\frac{2}{r}N_{r\quad\theta}^{(s)}} + {\frac{1}{r}\frac{\partial N_{\theta\theta}^{(s)}}{\partial\theta}} + \tau_{\theta}} = 0.}} & (2.9)\end{matrix}$The out-of-plane moment and force equilibrium equations are given by$\begin{matrix}{{{\frac{\partial M_{r}}{\partial r} + \frac{M_{r} - M_{\theta}}{r} + {\frac{1}{r}\frac{\partial M_{r\quad\theta}}{\partial\theta}} + Q_{r} - {\frac{h_{s}}{2}\tau_{r}}} = 0},{{\frac{\partial M_{r\quad\theta}}{\partial r} + {\frac{2}{r}M_{r\quad\theta}} + {\frac{1}{r}\frac{\partial M_{\theta}}{\partial\theta}} + Q_{\theta} - {\frac{h_{s}}{2}\tau_{\theta}}} = 0},} & (2.10) \\{{{\frac{\partial Q_{r}}{\partial r} + \frac{Q_{r}}{r} + {\frac{1}{r}\frac{\partial Q_{\theta}}{\partial\theta}}} = 0},} & (2.11)\end{matrix}$where Q_(r) and Q_(θ) are the shear forces normal to the neutral axis.The substitution of Eq. (2.7) into Eq. (2.9) yields the followinggoverning equations for u_(r) ^((s)) and u_(θ) ^((s)) (and τ)$\begin{matrix}{{{\frac{\partial^{2}u_{r}^{(f)}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial r}} - \frac{u_{r}^{(s)}}{r^{2}} + {\frac{1 - v_{s}}{2}\frac{1}{r^{2}}\frac{\partial^{2}u_{r}^{(s)}}{\partial\theta^{2}}} + {\frac{1 - v_{s}}{2}\frac{1}{r}\frac{\partial^{2}u_{\theta}^{(s)}}{{\partial r}{\partial\theta}}} - {\frac{3 - v_{s}}{2}\frac{1}{r^{2}}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} = {{- \frac{1 - v_{s}^{2}}{E_{s}h_{s}}}\tau_{r}}},{{{\frac{1 + v_{s}}{2}\frac{1}{r}\frac{\partial^{2}u_{r}^{(s)}}{{\partial r}{\partial\theta}}} + {\frac{3 - v_{s}}{2}\frac{1}{r^{2}}\frac{\partial u_{r}^{(s)}}{\partial\theta}} + {\frac{1 - v_{f}}{2}\left( {\frac{\partial^{2}u_{\theta}^{(s)}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial r}} - \frac{u_{\theta}^{(s)}}{r^{2}}} \right)} + {\frac{1}{r^{2}}\frac{\partial^{2}u_{\theta}^{(s)}}{\partial\theta^{2}}}} = {{- \frac{1 - v_{s}^{2}}{E_{s}h_{s}}}{\tau_{\theta}.}}}} & (2.12)\end{matrix}$Elimination of Q_(r) and Q_(θ) from Eqs. (2.10) and (2.11), inconjunction with Eq. (2.8), give the following governing equation for w(and τ) $\begin{matrix}{{{\nabla^{2}\left( {\nabla^{2}w} \right)} = {\frac{6\left( {1 - v_{s}^{2}} \right)}{E_{s}h_{s}^{2}}\left( {\frac{\partial\tau_{r}}{\partial r} + \frac{\tau_{r}}{r} + {\frac{1}{r}\frac{\partial\tau_{\theta}}{\partial\theta}}} \right)}},{{{where}\quad\nabla^{2}} = {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r^{2}}} + {\frac{1}{r^{2}}{\frac{\partial^{2}}{\partial\theta^{2}}.}}}}} & (2.13)\end{matrix}$The continuity of displacements across the thin film/substrate interfacerequires $\begin{matrix}{{u_{r}^{(f)} = {u_{r}^{(s)} - {\frac{h_{s}}{2}\frac{\partial w}{\partial r}}}},\quad{u_{\theta}^{(f)} = {u_{\theta}^{(s)} - {\frac{h_{s}}{2}\frac{1}{r}{\frac{\partial w}{\partial\theta}.}}}}} & (2.14)\end{matrix}$

Equations (2.4) and (2.12)-(2.14) constitute seven ordinary differentialequations for seven variables, namely u_(r) ^((f)), u_(θ) ^((f)), u_(r)^((s)), u_(θ) ^((s)), w, τ_(r) and τ_(θ).

We discuss in the following how to decouple these seven equations underthe limit h_(f)/h_(s)<<1 such that we can solve u_(r) ^((s)), u_(θ)^((s)) first, then w, followed by u_(r) ^((f)) and u_(θ) ^((f)), andfinally τ_(r), and τ_(θ).

(i) Elimination of τ, and τ_(θ) from force equilibrium equations (2.4)for the thin film and (2.12) for the substrate yields two equations foru_(r) ^((f)), u_(θ) ^((f)), u_(r) ^((s)) and u_(θ) ^((s)). Forh_(f)/h_(s)<1, u_(r) ^((f)) and u_(θ) ^((f)) disappear in these twoequations, which become the following governing equations for u_(r)^((s)) and u_(θ) ^((s)) only, $\begin{matrix}{{{\frac{\partial^{2}u_{r}^{(s)}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial r}} - \frac{u_{r}^{(s)}}{r^{2}} + {\frac{1 - v_{s}}{2}\frac{1}{r^{2}}\frac{\partial^{2}u_{r}^{(s)}}{\partial\theta^{2}}} + {\frac{1 + v_{s}}{2}\frac{1}{r}\frac{\partial^{2}u_{\theta}^{(s)}}{{\partial r}\quad{\partial\theta}}} - {\frac{3 - v_{s}}{2}\frac{1}{r^{2}}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} = {{\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{\partial ɛ^{m}}{\partial r}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},{{{\frac{1 + v_{s}}{2}\frac{1}{r}\frac{\partial^{2}u_{r}^{(s)}}{{\partial r}\quad{\partial\theta}}} + {\frac{3 - v_{s}}{2}\frac{1}{r^{2}}\frac{\partial u_{r}^{(s)}}{\partial\theta}} + {\frac{1 - v_{s}}{2}\left( {\frac{\partial^{2}u_{\theta}^{(s)}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial r}} - \frac{u_{\theta}^{(s)}}{r^{2}}} \right)} + {\frac{1}{r^{2}}\frac{\partial^{2}u_{r}^{(s)}}{\partial\theta^{2}}}} = {{\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{1}{r}\frac{\partial ɛ^{m}}{\partial\theta}} + {{O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}.}}}} & (2.15)\end{matrix}$The substrate displacements u_(r) ^((s)) and u_(θ) ^((s)) are on theorder ${of}\quad{\frac{h_{f}}{h_{s}}.}$

(ii) Elimination of u_(r) ^((f)) and u_(θ) ^((f)) from the continuitycondition (2.14) and equilibrium equation (2.4) for the thin film givesτ_(r) and τ_(θ) in terms of u_(r) ^((s)), u_(θ) ^((s)) and w (and ε^(m))

(iii) The substitution of the above τ, and τ_(θ) into the momentequilibrium equation (2.13) yields the governing equation for the normaldisplacement w. For h_(f)/h_(s)<<1, this governing equation takes theform $\begin{matrix}{{\nabla^{2}\left( {\nabla^{2}w} \right)} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}{{\nabla^{2}ɛ^{m}}.}}} & (2.16)\end{matrix}$This is a biharmonic equation which can be solved analytically. Thesubstrate displacement w is on the order of $\frac{h_{f}}{h_{s}}.$

(iv) The displacements u_(r) ^((f)) and u_(θ) ^((f)) in the thin filmare obtained from Eq. (2.14), and they are also on the same order$\frac{h_{f}}{h_{s}}$as u_(r) ^((s)), u_(θ) ^((s)) and w. The leading terms of the interfaceshear stresses τ, and τ₀ are then obtained from Eq. (2.4) as$\begin{matrix}{{\tau_{r} = {{- \frac{E_{f}h_{f}}{1 - v_{f}}}\frac{\partial ɛ^{m}}{\partial r}}},\quad{\tau_{\theta} = {{- \frac{E_{f}h_{f}}{1 - v_{f}}}\frac{1}{r}{\frac{\partial ɛ^{m}}{\partial\theta}.}}}} & (2.17)\end{matrix}$

These are remarkable results that hold regardless of boundary conditionsat the edge r=R. Therefore the interface shear stresses are proportionalto the gradients of misfit strain. For uniform misfit strain theinterface shear stresses vanish.

We expand the arbitrary non-uniform misfit strain distributionε^(m)(r,θ) to the Fourier series, $\begin{matrix}{{{ɛ^{m}\left( {r,\theta} \right)} = {{\sum\limits_{n = 0}^{\infty}{{ɛ_{c}^{m{(n)}}(r)}\cos\quad n\quad\theta}} + {\sum\limits_{n = 1}^{\infty}{{ɛ_{s}^{m{(n)}}(r)}\sin\quad n\quad\theta}}}},} & (2.18)\end{matrix}$

where${{ɛ_{c}^{m{(0)}}(r)} = {\frac{1}{2\quad\pi}{\int_{0}^{2\quad\pi}{{ɛ^{m}\left( {r,\theta} \right)}\quad{\mathbb{d}\theta}}}}},\quad\quad{{ɛ_{c}^{m{(n)}}(r)} = {\frac{1}{\quad\pi}{\int_{0}^{2\quad\pi}{{ɛ^{m}\left( {r,\theta} \right)}\cos\quad n\quad\theta\quad{\mathbb{d}\theta}\quad\left( {n \geq 1} \right)\quad{and}}}}}$${{ɛ_{s}^{m{(n)}}(r)} = {\frac{1}{\quad\pi}{\int_{0}^{2\quad\pi}{{ɛ^{m}\left( {r,\theta} \right)}\sin\quad n\quad\theta\quad{\mathbb{d}\theta}\quad{\left( {n \geq 1} \right)\quad.}}}}}\quad$Without losing generality, we focus on the cos nθ term here. Thecorresponding displacements and interface shear stresses can beexpressed asu _(r) ^((s)) =u _(r) ^((sn))(r)cos nθ, u _(θ) ^((s)) =u _(θ)^((sn))(r)sin nθ, w=w ^((n))(r)cos θ.   (2.19)

Equation (2.15) then provides two ordinary differential equations foru_(r) ^((sn)) and u₀ ^((sn)), which have the general solution$\begin{matrix}{{\begin{Bmatrix}u_{r}^{({sn})} \\u_{\theta}^{({sn})}\end{Bmatrix} = {{\begin{Bmatrix}{1 - v_{s} - {\frac{1 + v_{s}}{2}n}} \\{{\frac{1 + v_{s}}{2}n} + 2}\end{Bmatrix}\left\lbrack {{A_{0}r^{n + 1}} + {\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 + v_{s}}{E_{s}h_{s}}\frac{1}{4\left( {n + 1} \right)}r\quad ɛ_{c}^{m{(n)}}}} \right\rbrack} + {\begin{Bmatrix}1 \\1\end{Bmatrix}\left\langle {\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 + v_{s}}{E_{s}h_{s}}\frac{1}{4\left( {n + 1} \right)}\begin{Bmatrix}{{{- \left\lbrack {1 - v_{s} - {\frac{n}{2}\left( {1 + v_{s}} \right)}} \right\rbrack}r\quad ɛ_{c}^{m{(n)}}} +} \\{2\left( {1 - v_{s}} \right)\left( {n + 1} \right)\frac{1}{r^{n + 1}}{\int_{0}^{r}{\eta^{1 + n}ɛ_{c}^{m{(n)}}\quad{\mathbb{d}\eta}}}}\end{Bmatrix}} \right\rangle} - {\begin{Bmatrix}{1 - v_{s} + {\frac{1 + v_{s}}{2}n}} \\{{\frac{1 + v_{s}}{2}n} - 2}\end{Bmatrix}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 + v_{s}}{E_{s}h_{s}}\frac{1}{4\left( {n - 1} \right)}r\quad ɛ_{c}^{m{(n)}}} + {\begin{Bmatrix}{- 1} \\1\end{Bmatrix}\left\langle {{D_{0}r^{n - 1}} - {\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 + v_{s}}{E_{s}h_{s}}\frac{1}{4\left( {n - 1} \right)}\begin{Bmatrix}{{\left\lbrack {1 - v_{s} + {\frac{n}{2}\left( {1 + v_{s}} \right)}} \right\rbrack r\quad ɛ_{c}^{m{(n)}}} -} \\{2\left( {1 - v_{s}} \right)\left( {n - 1} \right)r^{n - 1}{\int_{0}^{R}{\eta^{1 - n}{ɛ_{c}^{m{(n)}}(\eta)}\quad{\mathbb{d}\eta}}}}\end{Bmatrix}}} \right\rangle} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},} & (2.20)\end{matrix}$where A₀ and D₀ are constants to be determined, and we have used thecondition that the displacements are finite at the center r=0.

The normal displacement w is obtained from the biharmonic equation(2.16) as $\begin{matrix}{{w^{(n)} = {{A_{1}r^{n + 2}} + {B_{1}r^{n}} + {\frac{3}{n}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}{\frac{E_{f}h_{f}}{1 - v_{f}}\left\lbrack \quad{{r^{n}{\int_{r}^{R}{\eta^{1 - n}{ɛ_{c}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}} + {r^{- n}{\int_{0}^{r}{\eta^{n - 1}{ɛ_{c}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}} + {O\left( \quad\frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},} & (2.21)\end{matrix}$where A₁ and B₁ are constants to be determined, and we have used thecondition that the displacement w is finite at the center r=0.

We now describe the corresponding boundary conditions for the analysis.The first two boundary conditions at the free edge r=R require that thenet forces vanish,N _(r) ^((f)) +N _(r) ^((s))=0 and N _(rθ) ^((f)) +N _(rθ) ^((s))=0 atr=R,   (3.1)where A₀ and D₀ are defined as: $\begin{matrix}{{A_{0} = {{\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}}\frac{1}{R^{{2n} + 2}}{\int_{0}^{R}{\eta^{n - 1}{ɛ_{c}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},{D_{0} = {{{- \frac{E_{f}h_{f}}{1 - v_{f}}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{n + 1}{2R^{2n}}{\int_{0}^{R}{\eta^{n + 1}{ɛ_{c}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}}} & (3.2)\end{matrix}$under the limit h_(f)/h_(s)<<1. The other two boundary conditions at thefree edge r=R are the vanishing of net moments, i.e., $\begin{matrix}{{{M_{r} - {\frac{h_{s}}{2}N_{r}^{(f)}}} = 0}{and}{{Q_{r} - {\frac{1}{r}\frac{\partial}{\partial\theta}\left( {M_{r\quad\theta} - {\frac{h_{s}}{2}N_{r\quad\theta}^{(f)}}} \right)}} = 0}{at}{{r = R},}} & (3.3)\end{matrix}$where A₁ and B₁ are defined as $\begin{matrix}{{A_{1} = {{{- \frac{3\left( {1 - v_{s}} \right)}{3 + v_{s}}}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\frac{1}{R^{{2n} + 2}}{\int_{0}^{R}{\eta^{n + 1}{ɛ_{c}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}} + {O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}}},{B_{1} = {{{- \frac{n + 1}{n}}R^{2}A_{1}} + {{O\left( \frac{h_{f}^{2}}{h_{s}^{2}} \right)}.}}}} & (3.4)\end{matrix}$

The boundary conditions can also be established from the knownvariational principle. The total potential energy in the thinfilm/substrate system with the free edge at r=R is $\begin{matrix}{{\Pi = {\int_{0}^{R}{r\quad{\mathbb{d}r}{\int_{0}^{2\pi}\quad{{\mathbb{d}\theta}{\int_{- \frac{h_{s}}{2}}^{\frac{h_{s}}{2} + h_{f}}{U{\mathbb{d}z}}}}}}}},} & (3.5)\end{matrix}$where U is the strain energy density which gives${\frac{\partial U}{\partial ɛ_{rr}} = \sigma_{rr}},{\frac{\partial U}{\partial ɛ_{\theta\theta}} = \sigma_{\theta\theta}}$and$\frac{\partial U}{\partial\gamma_{r\quad\theta}} = {\sigma_{r\quad\theta}.}$For constitutive relations in Eqs. (2.1) and (2.6), we obtain$\begin{matrix}{{U = {\frac{E}{2\left( {1 - v^{2}} \right)}\left\lbrack {ɛ_{rr}^{2} + ɛ_{\theta\theta}^{2} + {2v\quad ɛ_{rr}ɛ_{\theta\theta}} + {\frac{1 - v}{2}\gamma_{r\quad\theta}^{2}} - {2\left( {1 + v} \right){ɛ^{m}\left( {ɛ_{rr} + ɛ_{\theta\theta}} \right)}}} \right\rbrack}},} & (3.6)\end{matrix}$where E and v take their corresponding values in the thin film (i.e.,E_(f) and v_(f) for${\frac{h_{s}^{\bullet}}{2} + h_{f}} \geq z \geq \frac{h_{s}^{\bullet}}{2}$and in the substrate (i.e., E_(s) and v_(s) for$\left. {\frac{h_{s}^{\bullet}}{2} \geq z \geq {- \frac{h_{s}^{\bullet}}{2}}} \right).$For the displacement fields in Section 2 and the associated strainfields, the potential energy Π in Eq. (3.5) becomes a quadratic functionof parameters A₀, D₀/A₁ and B₁. The principle of minimum potentialenergy requires $\begin{matrix}{{{\frac{\partial\Pi}{\partial A_{0}} = 0},{\frac{\partial\Pi}{\partial D_{0}} = 0},{\frac{\partial\Pi}{\partial A_{1}} = 0}}{and}{\frac{\partial\Pi}{\partial B_{1}} = 0.}} & (3.7)\end{matrix}$It can be shown that, as expected in the limit h_(f)/h_(s)<<1, the abovefour conditions in Eq. (3.7) are equivalent to the vanishing of netforces in Eq. (3.1) and net moments in Eq. (3.3).

We now describe analysis of thin-film stresses and substrate curvatures.We provide the general solution that includes both cosine and sine termsin this section. The substrate curvatures are $\begin{matrix}{{\kappa_{rr} = \frac{\partial^{2}w}{\partial r^{2}}},{\kappa_{\theta\theta} = {{\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}}},{\kappa_{r\quad\theta} = {\frac{\partial}{\partial r}\left( {\frac{1}{r}\frac{\partial w}{\partial\theta}} \right)}}} & (4.1)\end{matrix}$The sum of substrate curvatures is related to the misfit strain by$\begin{matrix}\begin{matrix}{{\kappa_{rr} + \kappa_{\theta\theta}} = {{- 12}\quad\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}*}} \\{\begin{Bmatrix}{ɛ^{m} - {\frac{1 - v_{s}}{2}\left( {ɛ^{m} - \overset{\_}{ɛ^{m}}} \right)} + {\frac{1 - v_{s}^{2}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\left( {n + 1} \right)\quad\frac{r^{n}}{R^{{2\quad n} + 2}}}}}} \\\left\lbrack {{\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{n + 1}{ɛ_{c}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{n + 1}{ɛ_{s}^{m{(n)}}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack\end{Bmatrix},}\end{matrix} & (4.2)\end{matrix}$where$\overset{\_}{ɛ^{m}} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{{ɛ^{m}\left( {\eta,\varphi} \right)}{\mathbb{d}A}}}}}$is the average misfit strain over the entire area A of the thin film,dA=ηdηdφ, and ε^(m) is also related to ε_(c) ^(m(0)) by$\overset{\_}{ɛ^{m}} = {\frac{2}{R^{2}}{\int_{0}^{R}{{{\eta ɛ}_{c}^{m{(0)}}(\eta)}{{\mathbb{d}\eta}.}}}}$The difference between two curvatures, κ_(rr)−κ_(θθ), and the twistκ_(rθ) are given by $\begin{matrix}\begin{matrix}{{\kappa_{rr} - \kappa_{\theta\theta}} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}*}} \\{\begin{Bmatrix}{ɛ^{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad ɛ_{c}^{m{(0)}}{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\propto}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\quad\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}}} \\{\left( {{\cos\quad n\quad\theta\quad{\int_{0}^{R}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right) -} \\{{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\left( {{\cos\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{r}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}} -} \\{\sum\limits_{n = 1}^{\infty}{\left( {n - 1} \right){r^{n - 2}\left( {{\cos\quad n\quad\theta\quad{\int_{r}^{R}{\eta^{1 - n}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{r}^{R}{\eta^{1 - n}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}}}\end{Bmatrix},}\end{matrix} & (4.3) \\\begin{matrix}{\kappa_{r\quad\theta} = {3\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}*}} \\{\begin{Bmatrix}{\frac{1 - v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\quad\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}} \\{\left( {{\sin\quad n\quad\theta\quad{\int_{0}^{R}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right) +} \\{{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\left( {{\sin\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{0}^{r}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}} -} \\{\sum\limits_{n = 1}^{\infty}{\left( {n - 1} \right)\quad{r^{n - 2}\left( {{\sin\quad n\quad\theta\quad{\int_{r}^{R}{\eta^{1 - n}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{r}^{R}{\eta^{1 - n}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}}}\end{Bmatrix}.}\end{matrix} & (4.4)\end{matrix}$

The stresses in the thin film are obtained from Eq. (2.1). Specifically,the sum of stresses σ_(rr) ^((f))+σ_(θθ) ^((f)) is related to the misfitstrain by $\begin{matrix}{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{f}}{1 - v_{f}}{\left( {{- 2}\quad ɛ^{m}} \right).}}} & (4.5)\end{matrix}$The difference between stresses, σ_(rr) ^((f))−σ_(θθ) ^((f)), and shearstress σ_(rθ) ^((f)) are given by $\begin{matrix}\begin{matrix}{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {4\quad E_{f}\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}*}} \\{\begin{Bmatrix}{ɛ^{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad ɛ_{c}^{m{(0)}}{\mathbb{d}\eta}}}} - {\sum\limits_{n = 1}^{\infty}\frac{n + 1}{r^{n + 2}}}} \\{\left( {{\cos\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{r}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right) -} \\{{\sum\limits_{n = 1}^{\infty}{\left( {n - 1} \right){r^{n - 2}\left( {{\cos\quad n\quad\theta\quad{\int_{r}^{R}{\eta^{1 - n}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{r}^{R}{\eta^{1 - n}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}}} -} \\{\frac{v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}} \\\left( {{\cos\quad n\quad\theta\quad{\int_{0}^{R}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} + {\sin\quad n\quad\theta{\int_{0}^{R}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)\end{Bmatrix},}\end{matrix} & (4.6) \\\begin{matrix}{\sigma_{r\quad\theta}^{(f)} = {2\quad E_{f}\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}*}} \\{\begin{Bmatrix}{{- {\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\left( {{\sin\quad n\quad\theta\quad{\int_{0}^{r}{\eta^{n - 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{0}^{r}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}}} +} \\{{\sum\limits_{n = 1}^{\infty}{\left( {n - 1} \right)\quad{r^{n - 2}\left( {{\sin\quad n\quad\theta\quad{\int_{r}^{R}{\eta^{1 - n}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{r}^{R}{\eta^{1 - n}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)}}} +} \\{\frac{v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}} \\\left( {{\sin\quad n\quad\theta\quad{\int_{0}^{R}{\eta^{n + 1}ɛ_{c}^{m{(n)}}{\mathbb{d}\eta}}}} - {\cos\quad n\quad\theta{\int_{0}^{R}{\eta^{n + 1}ɛ_{s}^{m{(n)}}{\mathbb{d}\eta}}}}} \right)\end{Bmatrix}.}\end{matrix} & (4.7)\end{matrix}$

For uniform misfit strain distribution ε^(m)=constant, the interfaceshear stresses in Eq. (2.17) vanish. The curvatures in the substrateobtained from Eqs. (4.2)-(4.4) become${\kappa = {\kappa_{rr} = {\kappa_{\theta\theta} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}ɛ^{m}}}}},{\kappa_{r\quad\theta} = 0}$

The stresses in the thin film obtained from Eqs. (4.5)-(4.7) become${\sigma^{(f)} = {\sigma_{rr}^{(f)} = {\sigma_{\theta\theta}^{(f)} = {\frac{E_{f}}{1 - v_{f}}\left( {- ɛ^{m}} \right)}}}},{\sigma_{r\quad\theta}^{(f)} = 0}$For this special case only, both stress and curvature states becomeequi-biaxial. The elimination of misfit strain ε^(m) from the above twoequations yields a simple relation${\sigma^{(f)} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\kappa}},$which is exactly the Stoney formula, and it has been used to estimatethe thin-film stress σ^((f)) from the substrate curvature κ, if themisfit strain, stress and curvature are all constant and if the platesystem shape is spherical.

In the following, we extend such a relation for arbitrarynon-axisymmetric misfit strain distribution.

The stresses and curvatures are all given in terms of misfit strain inthe previous section. We extend the Stoney formula for arbitrarynon-uniform and non-axisymmetric misfit strain distribution in thissection by establishing the direct relation between the thin-filmstresses and substrate curvatures.

We first define the coefficients C_(n) and S_(n) related to thesubstrate curvatures by $\begin{matrix}{{C_{n} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\left( \frac{\eta}{R} \right)^{n}\cos\quad n\quad\varphi{\mathbb{d}A}}}}}},{S_{n} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\left( \frac{\eta}{R} \right)^{n}\sin\quad n\quad\varphi{\mathbb{d}A}}}}}},} & (5.1)\end{matrix}$where the integration is over the entire area A of the thin film, anddA=ηdηdφ. Since both the substrate curvatures and film stresses dependon the misfit strain ε^(m), elimination of misfit strain gives the filmstress in terms of substrate curvatures by $\begin{matrix}{{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\left\{ {{4\left( {\kappa_{rr} - \kappa_{\theta\theta}} \right)} - {\sum\limits_{n = 1}^{\infty}{{\left( {n + 1} \right)\left\lbrack {{n\left( \frac{r}{R} \right)}^{n} - {\left( {n - 1} \right)\left( \frac{r}{R} \right)^{n - 2}}} \right\rbrack}\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)}}} \right\}}},} & (5.2) \\{{\sigma_{r\quad\theta}^{(f)} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\left\{ {{4\kappa_{r\quad\theta}} + {\frac{1}{2}{\sum\limits_{n = 1}^{\infty}{{\left( {n + 1} \right)\left\lbrack {{n\left( \frac{r}{R} \right)}^{n} - {\left( {n - 1} \right)\left( \frac{r}{R} \right)^{n - 2}}} \right\rbrack}\left( {{C_{n}\sin\quad n\quad\theta} - {S_{n}\cos\quad n\quad\theta}} \right)}}}} \right\}}},} & (5.3) \\{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}h_{s}^{2}}{6\quad{h_{f}\left( {1 - v_{s}} \right)}}\begin{bmatrix}{\kappa_{rr} + \kappa_{\theta\theta} + {\frac{1 - v_{s}}{1 + v_{s}}\left( {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\quad\theta}}} \right)}} \\{{- \frac{1 - v_{s}}{1 + v_{s}}}{\sum\limits_{n = 1}^{\infty}{\left( {n + 1} \right)\left( \frac{r}{R} \right)^{n}\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)}}}\end{bmatrix}}},} & (5.4)\end{matrix}$where$\overset{\_}{\kappa_{rr} + \kappa_{\theta\quad\theta}} = {C_{0} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right){\mathbb{d}A}}}}}}$is the average curvature over entire area A of the thin film. Equations(5.2)-(5.4) provide direct relations between individual film stressesand substrate curvatures. It is important to note that stresses at apoint in the thin film depend not only on curvatures at the same point(local dependence), but also on the curvatures in the entire substrate(non-local dependence) via the coefficients C_(n) and S_(n).

The interface shear stresses τ_(r) and τ_(θ) can also be directlyrelated to substrate curvatures via $\begin{matrix}{{\tau_{r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\left\lbrack {{\frac{\partial}{\partial r}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)} - {\frac{1 - v_{s}}{2\quad R}{\sum\limits_{n = 1}^{\infty}{{n\left( {n + 1} \right)}\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)\quad\left( \frac{r}{R} \right)^{n - 1}}}}} \right\rbrack}},} & (5.5) \\{\tau_{\theta} = {{\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\left\lbrack {{\frac{1}{r}\frac{\partial}{\partial\theta}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)} - {\frac{1 - v_{s}}{2\quad R}{\sum\limits_{n = 1}^{\infty}{{n\left( {n + 1} \right)}\left( {{C_{n}\sin\quad n\quad\theta} - {S_{n}\cos\quad n\quad\theta}} \right)\quad\left( \frac{r}{R} \right)^{n - 1}}}}} \right\rbrack}.}} & (5.6)\end{matrix}$

This provides a way to estimate the interface shear stresses from thegradients of substrate curvatures. It also displays a non-localdependence via the coefficients C_(n) and S_(n).

Since interfacial shear stresses are responsible for promoting systemfailures through delamination of the thin film from the substrate, Eqs.(5.5) and (5.6) have particular significance. They show that suchstresses are related to the gradients of κ_(rr)+κ_(θθ) and not to itsmagnitude as might have been expected of a local, Stoney-likeformulation. Eqs. (5.5) and (5.6) can be implemented to provide an easyway of inferring these special interfacial shear stresses once thefull-field curvature information is available. As a result, themethodology also provides a way to evaluate the risk of and to mitigatesuch important forms of failure. For the special case of spatiallyconstant curvatures, the interfacial shear stresses vanish as is thecase for all Stoney-like formulations.

It can be shown that the relations between the film stresses andsubstrate curvatures given in the form of infinite series in Eqs.(5.2)-(5.4) can be equivalently expressed in the form of integration as$\begin{matrix}\begin{matrix}{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\left\{ {{4\left\{ {\kappa_{rr} - \kappa_{\theta\theta}} \right)} -} \right.}} \\{\left. {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\quad\frac{\frac{\eta}{R}{F_{minus}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}}}} \right\},}\end{matrix} & (5.7) \\\begin{matrix}{\sigma_{r\quad\theta}^{(f)} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\left\{ {{4\kappa_{r\quad\theta}} - {\frac{1}{2}\frac{1}{\pi\quad R^{2}}{\int{\int_{A}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)}}}} \right.}} \\{\left. {\frac{\frac{\eta}{R}{F_{shear}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}} \right\},}\end{matrix} & (5.8) \\\begin{matrix}{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}h_{s}^{2}}{6\quad{h_{f}\left( {1 - v_{f}} \right)}}*}} \\{\left\{ {\begin{matrix}{\kappa_{rr} + \kappa_{\theta\theta} + {\frac{1 - v_{s}}{1 + v_{s}}\left( {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\quad\theta}}} \right)} -} \\{\frac{1 - v_{s}}{1 + v_{s}}\frac{r}{\pi\quad R^{3}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\quad\frac{\frac{\eta}{R}{F_{plus}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}}}}}\end{matrix}{\mathbb{d}A}} \right\},}\end{matrix} & (5.9)\end{matrix}$where functions F_(min us), F_(shear) and F_(plus) are given byF _(minus)(r ₁,η₁,φ₁)=−r ₁ ²η₁(6+9η₁ ² +r ₁ ²η₁ ⁴)+r ₁(2+9η₁ ²+6r ₁ ²+6r₁ ²η₁ ⁴)cos φ₁−η₁(3+3r ₁ ²η₁ ²+2r ₁ ⁴η₁ ²)cos 2φ₁ +r ₁η₁ ² cos 3φ₁,F _(shear)(r ₁,η₁,φ₁)=r ₁(2+9η₁ ²−6r ₁ ²η₁ ²)sin φ₁−η₁(3+3r ₁ ²η₁ ²−2r ₁⁴η₁ ²)sin 2φ₁ +r ₁η₁ ² sin 3φ₁,F _(plus)(r ₁,η₁,φ₁)=2(1+2r ₁ ²η₁ ²)cos φ₁ −r ₁η₁ cos 2φ₁ −r ₁η₁(4+r ₁²η₁ ²).   (5.10)The interface shear stresses can also be related to substrate curvaturesvia integrals as $\begin{matrix}\begin{matrix}{\tau_{r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\left\{ {{\frac{\partial}{\partial r}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)} - \frac{1 - v_{s}}{\pi\quad R^{3}}} \right.}} \\{\left. {\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\quad\frac{\frac{\eta}{R}{F_{radial}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}}} \right\},}\end{matrix} & (5.11) \\\begin{matrix}{\tau_{\theta} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\left\{ {{\frac{1}{r}\frac{\partial}{\partial\theta}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)} - \frac{1 - v_{s}}{\pi\quad R^{3}}} \right.}} \\{\left. {\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\quad\frac{\frac{\eta}{R}{F_{circumferantial}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}}} \right\},}\end{matrix} & (5.12)\end{matrix}$whereF _(radial)(r ₁,η₁,φ₁)=(1+3r ₁ ²η₁ ²)cos φ₁ −r ₁η₁(3+r ₁ ²η₁ ² cos 2φ₁),F _(circumferantial)(r ₁,η₁,φ₁)=(1−3r ₁ ²η₁ ²)sin φ₁ +r ₁ ³η₁ ³ sin 2φ₁.  (5.13)

Eq. (5.4) also reduces to Stoney's result for the case of spatialcurvature uniformity. Indeed for this case, Eq. (5.4) reduces to:$\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right).}}} & (5.14)\end{matrix}$If in addition the curvature state is equi-biaxial (κ_(rr)=κ_(θθ)), asassumed by Stoney, the Stoney formulation is recovered while relation(5.2) furnishes σ_(rr)=σ_(θθ) (stress equi-biaxiality) as a specialcase.

Unlike Stoney's original analysis and some of its extensions, thepresent analysis shows that the dependence of film stresses on substratecurvatures is not generally “local.” Here the stress components at apoint on the film will, in general, depend on both the local value ofthe curvature components (at the same point) and on the value ofcurvatures of all other points on the plate system (non-localdependence). The more pronounced the curvature non-uniformities are, themore important such non-local effects become in accurately determiningfilm stresses from curvature measurements. This demonstrates thatanalyses methods based on Stoney's approach and its various extensionscannot handle the non-locality of the stress/curvature dependence andmay result in substantial stress prediction errors if such analyses areapplied locally in cases where spatial variations of system curvaturesand stresses are present.

The presence of non-local contributions in such relations also hasimplications regarding the nature of diagnostic methods needed toperform wafer-level film stress measurements. Notably the existence ofnon-local terms necessitates the use of full-field methods capable ofmeasuring curvature components over the entire surface of the platesystem (or wafer). Furthermore measurement of all independent componentsof the curvature field is necessary. This is because the stress state ata point depends on curvature contributions (from κ_(rr), κ_(θθ) andκ_(rθ)) from the entire plate surface.

Regarding the curvature-misfit strain [Eqs. (4.2)-(4.4)] andstress-misfit strain [Eqs. (4.5) -(4.7)] relations the following pointsare noteworthy. These relations also generally feature a dependence oflocal misfit strain ε^(m)(r,θ) which is “Stoney-like” as well as a“non-local” contribution from the misfit strain of other points on theplate system. Furthermore the stress and curvature states are alwaysnon-equibiaxial (i.e., σ_(rr) ^((f))≠σ_(θθ) ^((f)) and κ_(rr)≠κ_(θθ)) inthe presence of misfit strain non-uniformities. Only if ε^(m)=constantthese states become equi-biaxial, the “non-local” contributions vanishand Stoney's original results are recovered as a special case.

The existence of non-uniformities also results in the establishment ofshear stresses along the film/substrate interface. These stresses are ingeneral related to the derivatives of the first curvature invariantκ_(rr)+κ_(θθ) [Eqs. (5.11) and (5.12)]. In terms of misfit strain theseinterfacial shear stresses are also related to the gradients of themisfit strain distribution ε^(m)(r,θ). The occurrence of such stressesis ultimately related to spatial non-uniformities and as a result suchstresses vanish for the special case of uniform κ_(rr)+κ_(θθ) or ε^(m)considered by Stoney and its various extensions. Since film delaminationis a commonly encountered form of failure during wafer manufacturing,the ability to estimate the level and distribution of such stresses fromwafer-level metrology might prove to be invaluable in enhancing thereliability of such systems.

3. Island Structures and Effects on Stresses Under Non-UniformAxisymmetric Misfit Strain and Temperature

The techniques described in this section can be applied to obtain asimple stress-curvature relation in a system with a thin film and asubstrate of different radii. Such thin film and substrate structuresare commonly used in various applications such as integrated circuits,micro fabricated structures, MEMS devices, and large panels. Remarkablythe same simple stress-curvature relation still holds in presence ofmismatch between radii of the film and the substrate radii mismatch.

Stoney used a plate system composed of a stress bearing thin film, ofthickness h_(f), deposited on a relatively thick substrate, of thicknessh_(s), and derived a simple relation between the curvature, κ, of thesystem and the stress, σ^((f)), of the film as follows: $\begin{matrix}{\sigma^{(f)} = {\frac{E_{s}h_{s}^{2}\kappa}{6\quad{h_{f}\left( {1 - v_{s}} \right)}}.}} & (1.1)\end{matrix}$In the above the subscripts “f” and “s” denote the thin film andsubstrate, respectively, and E and v are the Young's modulus andPoisson's ratio. Equation (1.1) is called the Stoney formula, and it hasbeen extensively used in the literature to infer film stress changesfrom experimental measurement of system curvature changes.

Stoney's formula requires a number of assumptions given in thefollowing:

(i) Both the film thickness h_(f) and the substrate thickness h_(s) areuniform, the film and substrate have the same radius R, and h_(f)<<h<<R;

(ii) The strains and rotations of the plate system are infinitesimal;

(iii) Both the film and substrate are homogeneous, isotropic, andlinearly elastic;

(iv) The film stress states are in-plane isotropic or equi-biaxial (twoequal stress components in any two, mutually orthogonal in-planedirections) while the out-of-plane direct stress and all shear stressesvanish;

(v) The system's curvature components are equi-biaxial (two equal directcurvatures) while the twist curvature vanishes in all directions; and

(vi) All surviving stress and curvature components are spatiallyconstant over the plate system's surface, a situation which is oftenviolated in practice.

The analysis in this section is to extend the techniques described abovethat provide non-location contributions by relaxing the assumption (i)such that the thin film and substrate may have different radii. Toillustrate the technique, we consider an exemplary case of a thin filmand substrate with different radii subjected to arbitrary, radiallysymmetric misfit strain field ε_(m)(r) in the thin film or temperaturefield T(r) in the thin film and substrate. Here the misfit strain refersto the intrinsic strain in thin film that is not associated with thestress. The examples described below relate film stresses and systemcurvatures to the misfit strain (or temperature) distribution and derivea relation between the film stresses and the system curvatures thatwould allow for the accurate experimental inference of film stress fromfull-field and real-time curvature measurements.

3.1 Nonuniform Misfit Strain

FIG. 1G shows a circular thin film of radius R_(f) is deposited on asubstrate of a larger radius R_(s)(>R_(f)). The film thickness h_(f) ismuch less than the substrate thickness h_(s), i.e., h_(f)<<h_(s). TheYoung's modulus and Poisson's ratio of the film and substrate aredenoted by E_(f),v_(f),E_(s) and v_(s), respectively. As an example,assume that the thin film is subject to axisymmetric misfit straindistribution ε_(m)(r), where r is the radial coordinate. Cylindricalcoordinates (r,θ,z) are used in FIG. 1G for this axisymmetric problem.

The thin film and substrate are modeled as a membrane and a plate,respectively, since the film is very thin and cannot be subjected tobending. Let u_(f) and u_(s) denote the radial displacements in the thinfilm and substrate (at the neutral axis). The strains are${ɛ_{rr} = {{\frac{\mathbb{d}u}{\mathbb{d}r}\quad{and}\quad ɛ_{\theta\theta}} = \frac{u}{r}}},$where u is u_(f) in the thin film and u_(s) in the substrate. The axialforces stresses in the film and substrate can be obtained from thelinear elastic constitutive model as $\begin{matrix}{{N_{r} = {\frac{E\quad h}{1 - v^{2}}\left\lbrack {\frac{\mathbb{d}u}{\mathbb{d}r} + {v\quad\frac{u}{r}} - {\left( {1 + v} \right)ɛ_{misfit}}} \right\rbrack}},{N_{\theta} = {\frac{E\quad h}{1 - v^{2}}\left\lbrack {{v\frac{\mathbb{d}u}{\mathbb{d}r}} + \frac{u}{r} - {\left( {1 + v} \right)ɛ_{misfit}}} \right\rbrack}},} & \left( {2.1{.1}} \right)\end{matrix}$where E, v., h and ε_(misfit) are E_(f), v_(f), h_(f) and ε_(m) in thethin film and E_(s), v_(s), h_(s) and 0 in the substrate.

The shear stress traction σ_(rz) at the film/substrate interface isdenoted by τ(r). The normal stress traction σ_(zz) vanishes because thethin film cannot be subjected to bending. The equilibrium of forcesrequires $\begin{matrix}{{{\frac{\mathbb{d}N_{r}}{\mathbb{d}r} + {\frac{N_{r} - N_{\theta}}{r} \mp \tau}} = 0},} & \left( {2.1{.2}} \right)\end{matrix}$where −τ and +τ are for the thin film and substrate within the filmportion (r≦R_(f)), respectively, and τ vanishes for the substrateoutside the film (R_(f)<r≦R_(s)). The substitution of (2.1.1) into(2.1.2) yields the following governing equations for u_(f), u_(s) and τ$\begin{matrix}{{{\frac{\mathbb{d}^{2}u_{f}}{\mathbb{d}r^{2}} + {\frac{1}{r}\frac{\mathbb{d}u_{f}}{\mathbb{d}r}} - \frac{u_{f}}{r^{2}}} = {{{\frac{1 - v_{f}^{2}}{E_{f}h_{f}}\tau} + {\left( {1 + v_{f}} \right)\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}\quad{for}\quad r}} \leq R_{f}}},} & \left( {2.1{.3}} \right) \\{{\frac{\mathbb{d}^{2}u_{s}}{\mathbb{d}r^{2}} + {\frac{1}{r}\frac{\mathbb{d}u_{s}}{\mathbb{d}r}} - \frac{u_{s}}{r^{2}}} = \left\{ \begin{matrix}{{- \frac{v_{s}^{2}}{E_{s}h_{s}}}{\tau.}} & {{{for}\quad r} \leq R_{f}} \\0 & {{{for}\quad R_{f}} < r \leq {R_{s}.}}\end{matrix} \right.} & \left( {2.1{.4}} \right)\end{matrix}$Let w denote the lateral displacement of the substrate in the normal (z)direction. The bending moments in the substrate are given in terms of wby $\begin{matrix}{{M_{r} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\left( {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} + {\frac{v_{s}}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right)}},{M_{\theta} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}{\left( {{v_{s}\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} + {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right).}}}} & \left( {2.1{.5}} \right)\end{matrix}$The out-of-plane force and moment equilibrium equations are$\begin{matrix}{{{\frac{\mathbb{d}M_{r}}{\mathbb{d}r} + \frac{M_{r} - M_{\theta}}{r} + Q - {\frac{h_{s}}{2}\tau}} = 0},{{\frac{\mathbb{d}Q}{\mathbb{d}r} + \frac{Q}{r}} = 0},} & \left( {2.1{.6}} \right)\end{matrix}$where Q is the shear force normal to the neutral axis,$\frac{h_{s}}{2}\tau$is the contribution from the shear stress τ at the film/substrateinterface within the film portion (r≦R_(f)), and it vanishes for thesubstrate outside the film (R_(f)<r≦R_(s)). The second equation in(2.1.6) and the requirement of finite Q at the center r=0 give Q=0 inthe entire substrate. The substitution of (2.1.5) into the firstequation in (2.6) then gives the following governing equation for w andτ $\begin{matrix}{{\frac{\mathbb{d}^{3}w}{\mathbb{d}r^{3}} + {\frac{1}{r}\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} - {\frac{1}{r^{2}}\frac{\mathbb{d}w}{\mathbb{d}r}}} = \left\{ \begin{matrix}{\frac{6\left( {1 - v_{s}^{2}} \right)}{E_{s}h_{s}^{2}}\tau} & {{{for}\quad r} \leq R_{f}} \\0 & {{{for}\quad R_{f}} < r \leq {R_{s}.}}\end{matrix} \right.} & \left( {2.1{.7}} \right)\end{matrix}$

The continuity of displacement across the thin film/substrate interfacerequires $\begin{matrix}{u_{f} = {{u_{s} - {\frac{h_{s}}{2}\frac{\mathbb{d}w}{\mathbb{d}r}\quad{for}\quad r}} \leq {R_{f}.}}} & \left( {2.1{.8}} \right)\end{matrix}$

Equations (2.1.3), (2.1.4), (2.1.7) and (2.1.8) constitute four ordinarydifferential equations for u_(f), u_(s), w and τ within the film portion(r≦R_(f)). Outside the film (R_(f)<r≦R_(s)) (2.1.4) and (2.1.7) governu_(s) and w.

The shear stress at the film/substrate interface can be obtained byeliminating u_(f), u_(s) and w from these four equations as$\begin{matrix}{{\tau = {{- \frac{E_{f}h_{f}}{1 - v_{f}}}{\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}\left\lbrack {1 + {O\left( \frac{h_{f}}{h_{s}} \right)}} \right\rbrack}}},} & \left( {2.1{.9}} \right)\end{matrix}$where the higher-order terms for h_(f)/h_(s)<<1 are neglected. This is aremarkable result that the interface shear stress is proportional to thegradient of misfit strain. Equations (2.1.4) and (2.1.7) can then besolved analytically as $\begin{matrix}{\frac{\mathbb{d}w}{\mathbb{d}r} = \left\{ \begin{matrix}{{{{- 6}\quad\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\frac{1}{r}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{B_{1}}{2}r\quad{for}\quad r}} \leq R_{f}} \\{{{{A_{1}r} + {\frac{C_{1}}{r}\quad{for}\quad R_{f}}} < r \leq R_{s}},}\end{matrix} \right.} & \left( {2.1{.10}} \right) \\{\quad{u_{s} = \left\{ \begin{matrix}{\quad{{{\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{1}{r}{\int_{0}^{r}{\eta\quad{ɛ_{m}(\eta)}{\mathbb{d}\eta}}}} + {\frac{B_{2}}{2}r\quad{for}\quad r}} \leq R_{f}}} \\{{{{A_{2}r} + {\frac{C_{2}}{r}\quad{for}\quad R_{f}}} < r \leq R_{s}},}\end{matrix} \right.}} & \left( {2.1{.11}} \right)\end{matrix}$where only the leading terms for h_(f)/h_(s)<<1 are shown, and B₁, A₁,C₁, B₂, A₂ and C₂ are constants to be determined by the continuityconditions across the edge of thin film (r=R_(f)) and boundaryconditions at the edge of substrate (r=R_(s)) to be given in the nextsection. The displacement u_(f) in the thin film can be obtained frominterface continuity condition in Eq. (2.1.8).3.1.1 Continuity Conditions and Boundary Conditions

The displacement continuity conditions at the edge of thin film require$\begin{matrix}{{\left\lbrack \frac{\mathbb{d}w}{\mathbb{d}r} \right\rbrack_{r = R_{f}} = {{0\quad{{and}\quad\left\lbrack u_{s} \right\rbrack}_{r = R_{f}}} = 0}},} & \left( {2.2{.1}} \right)\end{matrix}$where [ . . . ] stands for the jump. The jump of the axial force insubstrate [N_(r) ^((s))] is related to the axial force in thin filmN_(r) ^((f)) byN _(r) ^((f))|_(r=R) _(f) −|[N _(r) ^((s))]_(r=R) _(f) =0   (2.2.2)such that the net external force vanishes. Similarly, the jump of thebending moment in substrate [M_(r)] is related to the bending momentproduced by the axial force in thin film with respect to the neutralaxis of substrate by $\begin{matrix}{\left. {{- \left\lbrack M_{r} \right\rbrack_{r = R_{f}}} - {\frac{h_{s} + h_{f}}{2}N_{r}^{(f)}}} \right|_{r = R_{f}} = 0} & \left( {2.2{.3}} \right)\end{matrix}$such that the net external moment vanishes.

The traction-free boundary conditions at the edge of substrate areN _(r) ^((s))|_(r-R) _(s) =0 and M _(r)|_(r=R) _(s) =0   (2.2.4)Equations (2.2.1)-(2.2.4) are 6 linear algebraic equations to determineB₁, A₁, C₁, B₂, A₂ and C₂. The displacements in (2.1.10), (2.1.11) and(2.1.8) are then given by $\begin{matrix}{\frac{\mathbb{d}w}{\mathbb{d}r} = \left\{ \begin{matrix}{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}r}} \right\rbrack}} & \quad & \begin{matrix}{for} & {r \leq R_{f}}\end{matrix} \\{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{{- \frac{\overset{\_}{ɛ_{m}}}{2}}\frac{R_{s}^{2}}{r}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}r}} \right\rbrack}} & \quad & \begin{matrix}{for} & {R_{f} < r \leq R_{s}}\end{matrix}\end{matrix} \right.} & \left( {2.2{.5}} \right) \\{u_{s} = \left\{ \begin{matrix}{\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\left\lbrack {{\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}{\frac{\overset{\_}{ɛ_{m}}}{2} \cdot r}}} \right\rbrack}} & \quad & \begin{matrix}{for} & {r \leq R_{f}}\end{matrix} \\{\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\left\lbrack {{\frac{\overset{\_}{ɛ_{m}}}{2}\frac{R_{s}^{2}}{r}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}r}} \right\rbrack}} & \quad & \begin{matrix}{for} & {{R_{f} < r \leq R_{s}},}\end{matrix}\end{matrix} \right.} & \left( {2.2{.6}} \right) \\{{u_{f} = {{4u_{s}\quad{for}\quad r} \leq R_{f}}},} & \left( {2.2{.7}} \right)\end{matrix}$where$\overset{\_}{ɛ_{m}} = {{\frac{2}{R_{s}^{2}}{\int_{0}^{R_{f}}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} = \frac{\underset{A_{film}}{\int\int}ɛ_{m}{\mathbb{d}A}}{A_{substrate}}}$is the average misfit strain, i.e., the misfit strain of the thin filmaveraged over the entire substrate.3.1.2. Thin Film Stresses and Substrate Curvatures

The substrate curvatures can be obtained from the displacement w as$\begin{matrix}{\kappa_{rr} = {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} = \left\{ {{\begin{matrix}{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\begin{bmatrix}{ɛ_{m} - {\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +} \\{\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}}\end{bmatrix}}} & {{{for}\quad r} \leq R_{f}} & \quad \\{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\frac{\overset{\_}{ɛ_{m}}}{2}\frac{R_{s}^{2}}{r^{2}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}}} \right\rbrack}} & {{{{for}\quad R_{f}} < r \leq R_{s}},} & \quad\end{matrix}\kappa_{\theta\theta}} = {{\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}} = \left\{ \begin{matrix}{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\begin{bmatrix}{{\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +} \\{\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}}\end{bmatrix}}} & {{{for}\quad r} \leq R_{f}} & \quad \\{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\frac{\overset{\_}{ɛ_{m}}}{2}\frac{R_{s}^{2}}{r^{2}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}}} \right\rbrack}} & {{{for}\quad R_{f}} < r \leq {R_{s}.}} & \quad\end{matrix} \right.}} \right.}} & \left( {2.3{.1}} \right)\end{matrix}$The circumferential curvature κ_(θθ) is continuous across the edge ofthin film (r=R_(f)), but the radial curvature has a jump. The sum ofthese two curvatures is $\begin{matrix}{{\kappa_{rr} + \kappa_{\theta\theta}} = \left\{ \begin{matrix}{{- 12}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}}{E_{s}h_{s}^{2}}\left\lbrack {ɛ_{m} - {\frac{1 - v_{s}}{2}\left( {ɛ_{m} - \overset{\_}{ɛ_{m}}} \right)}} \right\rbrack}} & {{{for}\quad r} \leq R_{f}} & \quad \\{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{\left( {1 - v_{s}} \right)^{2}}{E_{s}h_{s}^{2}}\overset{\_}{ɛ_{m}}} & {{{{for}\quad R_{f}} < r \leq R_{s}},} & \quad\end{matrix} \right.} & \left( {2.3{.2}} \right)\end{matrix}$where the first term on the right hand side of the first linecorresponds to the local misfit strain ε_(m), while the second termgives the deviation from the local misfit strain and is proportional tothe difference between the local misfit strain and the average misfitstrain ε_(m)− ε_(m) . The difference between two curvatures in (2.3.1)is $\begin{matrix}{{\kappa_{rr} + \kappa_{\theta\theta}} = \left\{ \begin{matrix}{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {ɛ_{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}}} \right\rbrack}} & {{{for}\quad r} \leq R_{f}} & \quad \\{{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{\left( {1 - v_{s}} \right)^{2}}{E_{s}h_{s}^{2}}\frac{R_{s}^{2}}{r^{2}}\overset{\_}{ɛ_{m}}} & {{{for}\quad R_{f}} < r \leq {R_{s}.}} & \quad\end{matrix} \right.} & \left( {2.3{.3}} \right)\end{matrix}$

The stresses in the thin film are obtained from (2.1.1), $\begin{matrix}{{\sigma_{rr}^{(f)} = {\frac{E_{f}}{1 - v_{f}}\left\{ {{- ɛ_{m}} + {4\frac{E_{f}h_{f}}{1 - v_{f}^{2}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\begin{bmatrix}{ɛ_{m} - {\left( {1 - v_{f}} \right)\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +} \\{\frac{1 + v_{f}}{2}\frac{1 - v_{s}}{1 + v_{s}}\overset{\_}{ɛ_{m}}}\end{bmatrix}}}} \right\}}}{\sigma_{\theta\theta}^{(f)} = {\frac{E_{f}}{1 - v_{f}}{\left\{ {{- ɛ_{m}} + {4\frac{E_{f}h_{f}}{1 - v_{f}^{2}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\begin{bmatrix}{{v_{f}ɛ_{m}} + {\left( {1 - v_{f}} \right)\frac{1}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} +} \\{\frac{1 + v_{f}}{2}\frac{1 - v_{s}}{1 + v_{s}}\overset{\_}{ɛ_{m}}}\end{bmatrix}}}} \right\}.}}}} & \left( {2.3{.4}} \right)\end{matrix}$The sum and difference of these stresses have the following simpleexpressions $\begin{matrix}{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{f}}{1 - v_{f}}\left( {{- 2}ɛ_{m}} \right)}}{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {4E_{f}\frac{E_{f}h_{f}}{1 - v_{f}^{2}}{{\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\left\lbrack {ɛ_{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}n}}}}} \right\rbrack}.}}}} & \left( {2.3{.5}} \right)\end{matrix}$

For uniform misfit strain ε_(m)=constant, the substrate curvaturesobtained from (2.3.1)-(2.3.3) become $\begin{matrix}{\kappa = {\kappa_{rr} = {\kappa_{\theta\theta} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}}{E_{s}h_{s}^{2}}\left\lbrack {1 - {\frac{1 - v_{s}}{2}\left( {1 - \frac{R_{f}^{2}}{R_{s}^{2}}} \right)}} \right\rbrack}ɛ_{m}}}}} & {{{{for}\quad r} \leq R_{f}},} & \quad\end{matrix}$ $\left\{ \begin{matrix}\begin{matrix}{\kappa_{rr} = {3\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}}{E_{s}h_{s}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\frac{R_{f}^{2}}{r^{2}}} - {\left( {1 - v_{s}} \right)\frac{R_{f}^{2}}{R_{s}^{2}}}} \right\rbrack}ɛ_{m}}} \\{\kappa_{rr} = {{- 3}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}}{E_{s}h_{s}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\frac{R_{f}^{2}}{r^{2}}} - {\left( {1 - v_{s}} \right)\frac{R_{f}^{2}}{R_{s}^{2}}}} \right\rbrack}ɛ_{m}}}\end{matrix} & {{{for}\quad R_{j}} < r \leq {R_{s}.}} & \quad\end{matrix} \right.$The curvature state is constant and equi-biaxial only within filmportion r≦R_(f). It can be easily verified that the circumferentialcurvature is continuous across the edge of thin film, but the radialcurvature has a jump. The thin film stresses from (2.3.5) become$\sigma = {\sigma_{rr}^{(f)} = {\sigma_{\theta\theta}^{(f)} = {\frac{E_{f}}{1 - v_{f}}{\left( {- ɛ_{m}} \right).}}}}$For this special case only, the stress state becomes equi-biaxial.Elimination of misfit strain ε_(m) from the above two equations yields asimple relation${\sigma = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right){h_{f}\left\lbrack {1 - {\frac{1 - v_{s}}{2}\left( {1 - \frac{R_{f}^{2}}{R_{s}^{2}}} \right)}} \right\rbrack}}\kappa}},$where κ is the constant curvature within the film portion r≦R_(f). Forthe thin film and substrate of same radii, the above relationdegenerates Stoney's formula in Eq. (1.1) which has been used toestimate the thin-film stress a from the substrate curvature κ, if themisfit strain, stress and curvature are all constants, and if the platesystem shape is spherical. For the thin film and substrate of differentradii, the radius effect clearly comes into play. In the following, weestablish a simple relation between film stress and substrate curvaturefor non-uniform misfit strain distribution.3.1.3. Extension of Stoney Formula for a Non-Uniform Misfit StrainDistribution

We extend the Stoney formula by establishing the direct relation betweenthe thin-film stresses and substrate curvatures for the thin film andsubstrate of different radii subject to a non-uniform misfitdistribution. Both κ_(rr)−κ_(θθ) in Eq. (2.3.3) and σ_(rr) ^((f))−σ_(θθ)^((f)) in Eq. (2.3.5) are proportional to${ɛ_{m}(r)} - {\frac{2}{r^{2}}{\int\limits_{0}^{r}{{{\eta ɛ}_{m}(\eta)}{{\mathbb{d}\eta}.}}}}$Therefore, elimination of misfit strain gives the difference σ_(rr)^((f))−σ_(θθ) ^((f)) in thin-film stresses directly proportional to thedifference κ_(rr)−κ_(θθ) in substrate curvatures, $\begin{matrix}{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {{- \frac{2E_{f}h_{s}}{3\left( {1 + v_{f}} \right)}}{\left( {\kappa_{rr} - \kappa_{\theta\theta}} \right).}}} & \left( {2.4{.1}} \right)\end{matrix}$This relation is independent of the thin film and substrate radii, andis identical to its counterpart for the thin film and substrate with thesame radii subjected to nonuniform misfit strain. The above relationclearly shows that the radial and circumferential stress components willbe equal only if the equivalent curvature components are also equal.

We now focus on the sum of thin-film stresses σ_(rr) ^((f))+σ_(θθ)^((f)) and sum of substrate curvatures κ_(rr)+κ_(θθ). We define theaverage substrate curvature κ_(rr)+κ_(θθ) as $\begin{matrix}\begin{matrix}{\overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}} = {\frac{1}{\quad A_{\quad{substrate}}}\underset{\quad A_{\quad{substrate}}}{\int\int}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)\eta{\mathbb{d}\eta}{\mathbb{d}\theta}}} \\{{= {\frac{2}{\quad R_{\quad s}^{\quad 2}}{\int\limits_{0}^{\quad R_{\quad s}}{\eta\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right){\mathbb{d}\eta}}}}},}\end{matrix} & \left( {2.4{.2}} \right)\end{matrix}$where the integration is over the entire area A_(substrate) of thesubstrate. The average substrate curvature can be related to the averagemisfit strain ε_(m) by averaging both sides of Eq. (2.3.3), i.e.,$\begin{matrix}{\overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}} = {12\frac{E_{f}h_{\quad f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}{\left( {- \overset{\_}{ɛ_{m}}} \right).}}} & \left( {2.4{.3}} \right)\end{matrix}$Elimination of misfit strain ε_(m) and average misfit strain ε_(m) givesthe sum of thin-film stresses in terms of curvatures as $\begin{matrix}{{\sigma_{rr} + \sigma_{\theta\theta}} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}{\left\{ {\kappa_{rr} + \kappa_{\theta\theta} + {\frac{1 - v_{s}}{1 + v_{s}}\left\lbrack {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}} \right\rbrack}} \right\}.}}} & \left( {2.4{.4}} \right)\end{matrix}$The above equation is once again independent of the thin film andsubstrate radii, and is identical to its counterpart for the thin filmand substrate with the same radii subjected to nonuniform misfit strain.Equations (2.4.4) and (2.4.1) provide direct relations between thin-filmstresses and substrate curvatures. Stresses at a point in the thin filmdepend not only on curvatures at the same point (local dependence), butalso on the average curvature in the entire substrate (non-localdependence).

The interface stress τ(r) given in Eq. (2.1.9) can also be directlyrelated to substrate curvatures via $\begin{matrix}{\tau = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\frac{\mathbb{d}}{\mathbb{d}r}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right).}}} & \left( {2.4{.5}} \right)\end{matrix}$This provides a remarkably simple way to estimate the interface shearstress from the radial gradient of the sum of two substrate curvatures.Equation (2.4.5) is independent of the thin film and substrate radii,and is identical to its counterpart for the thin film and substrate withthe same radii subjected to nonuniform misfit strain.

Since interfacial shear stresses are responsible for promoting systemfailures through delamination of the thin film from the substrate, Eq.(2.4.5) has particular significance. It shows that such stresses areproportional to the radial gradient of κ_(rr)+κ_(θθ) and not to itsmagnitude as might have been expected of a local, Stoney-likeformulation. Eq. (2.4.5) can be used to provide an easy way of inferringthese special interfacial shear stresses once the full-field curvatureinformation is available. As a result, the methodology also provides away to evaluate the risk of and to mitigate such important forms offailure. For the special case of spatially constant curvatures, thisinterfacial shear stress r vanishes and the above equation becomes aStoney-like equation.

3.2 Nonuniform Temperature

We now consider the thin film and substrate of radii R_(f) andR_(s)(>R_(f)) subject to nonuniform temperature change T(r). Once again,an axisymmetric distribution is considered as an example.

The linear elastic constitutive model (2.1.1) still holds except thatthe misfit strain ε_(misfit) is replaced by α_(f)T for the thin film andα_(s)T for the substrate, where α_(f) and α_(s) are the coefficients ofthermal expansion. The equilibrium equations (2.1.2) and (2.1.6),moment-curvature relation (2.1.5), displacement continuity (2.1.8)across the thin film/substrate interface, and continuity and boundaryconditions in Section 2.2 also hold.

The shear stress at the film/substrate interface is given by$\begin{matrix}{\tau = {{\frac{E_{f}h_{f}}{1 - v_{f}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack}{\frac{\mathbb{d}T}{\mathbb{d}r}.}}} & (3.1)\end{matrix}$This is a remarkable result that the interface shear stress isproportional to the gradient of temperature change. The displacementsare given by $\begin{matrix}{\frac{\mathbb{d}w}{\mathbb{d}r} = \left\{ \begin{matrix}{{\frac{6E_{f}h_{f}}{1 - v_{f}^{2}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack}\frac{1}{r}{\int\limits_{0}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}} + {\frac{B_{1}}{2}r}} & {for} & {r \leq R_{f}} \\{\frac{3E_{f}h_{f}}{2\left( {1 - v_{f}} \right)}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\{ {{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {2\alpha_{f}}} \right\rbrack\overset{\_}{T_{f}}} + {\left( {1 - v_{s}} \right)\alpha_{s}\overset{\_}{T_{s}}}} \right\}\left( {{\frac{1 - v_{s}}{1 + v_{s}}\frac{R_{f}^{2}}{R_{s}^{2}}r} + \frac{R_{f}^{2}}{r}} \right)} & {for} & {R_{f} < r \leq R_{s}}\end{matrix} \right.} & (3.2) \\{{u_{s} = {{{\left( {1 + v_{s}} \right)\alpha_{s}\frac{1}{r}{\int_{0}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}} + {\frac{1}{2}\left( {1 - v_{s}} \right)\alpha_{s}\overset{\_}{T_{s}}\quad{for}\quad r}} \leq R_{s}}},} & (3.3) \\{{u_{f} = {{u_{s}\quad{for}\quad r} \leq R_{f}}},} & (3.4)\end{matrix}$where${\overset{\_}{T}}_{f} = {{\frac{1}{A_{film}}\underset{A_{film}}{\int\int}T\quad\eta{\mathbb{d}\eta}{\mathbb{d}\theta}} = {\frac{2}{R_{f}^{2}}{\int_{0}^{R_{f}}{\eta\quad T{\mathbb{d}\eta}}}}}$and${\overset{\_}{T}}_{f} = {{\frac{1}{A_{substrate}}\underset{A_{substrate}}{\int\int}T\quad\eta{\mathbb{d}\eta}{\mathbb{d}\theta}} = {\frac{2}{R_{s}^{2}}{\int_{0}^{R_{s}}{\eta\quad T{\mathbb{d}\eta}}}}}$are the average temperatures in the film and substrate, respectively,and $\begin{matrix}{\frac{B_{1}}{2} = {\frac{3E_{f}h_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}{\begin{Bmatrix}{\frac{1 + v_{f}}{2}\left( {{\frac{1 - v_{s}}{1 + v_{s}}\frac{R_{f}^{2}}{R_{s}^{2}}} + 1} \right)} \\{\begin{Bmatrix}{\left\lbrack {{\left( {1 + v_{\quad s}} \right)\alpha_{\quad s}} - {2\quad\alpha_{\quad f}}} \right\rbrack\overset{\_}{\quad T_{f}}} \\{{+ \left( {1 - v_{\quad s}} \right)}\alpha_{s}\quad T_{s}}\end{Bmatrix} -} \\{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack\overset{\_}{T_{f}}}\end{Bmatrix}.}}} & (3.5)\end{matrix}$

The sum of two substrate curvatures is $\begin{matrix}{{\kappa_{rr} + \kappa_{\theta\theta}} = \left\{ \begin{matrix}{{\frac{6E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack}T} + B_{1}} & {for} & {r \leq R_{f}} \\{\frac{3E_{f}h_{f}}{1 - v_{f}}\frac{\left( {1 - v_{s}} \right)^{2}}{E_{s}h_{s}^{2}}\frac{R_{f}^{2}}{R_{s}^{2}}\left\{ {{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {2\alpha_{f}}} \right\rbrack\overset{\_}{T_{f}}} + {\left( {1 - v_{s}} \right)\alpha_{s}\overset{\_}{T_{s}}}} \right\}} & {for} & {{R_{f} < r \leq R_{s}},}\end{matrix} \right.} & (3.6)\end{matrix}$and the difference between two substrate curvatures is $\begin{matrix}{{\kappa_{rr} - \kappa_{\theta\theta}} = \left\{ \begin{matrix}{\frac{6E_{f}h_{f}}{1 - v_{f}^{2}}{{\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}} \right\rbrack}\left\lbrack {T - {\frac{1}{r^{2}}{\int_{0}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}} & {for} & {r \leq R_{f}} \\{{- \frac{3E_{f}h_{f}}{1 - v_{f}}}\frac{\left( {1 - v_{s}} \right)^{2}}{E_{s}h_{s}^{2}}\frac{R_{f}^{2}}{R_{s}^{2}}\left\{ {{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {2\alpha_{f}}} \right\rbrack\overset{\_}{T_{f}}} + {\left( {1 - v_{s}} \right)\alpha_{s}\overset{\_}{T_{s}}}} \right\}\frac{R_{f}^{2}}{r^{2}}} & {for} & {R_{f} < r \leq R_{s}}\end{matrix} \right.} & (3.7)\end{matrix}$Similarly, the sum and difference of thin-film stresses are given by$\begin{matrix}{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{f}}{1 - v_{f}}\left\{ {{\left\lbrack {{\left( {1 + v_{s}} \right)\alpha_{s}} - {2\alpha_{f}}} \right\rbrack T} + {\left( {1 - v_{s}} \right)\alpha_{s}\overset{\_}{T_{s}}}} \right\}}}{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {\frac{E_{f}}{1 + v_{f}}\left( {1 + v_{s}} \right){{\alpha_{s}\left\lbrack {T - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\quad{T(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}.}}}} & (3.8)\end{matrix}$

Elimination of temperature change gives the difference σ_(rr)^((f))−σ_(θθ) ^((f)) in thin-film stresses directly proportional to thedifference κ_(rr)−κ_(θθ) in substrate curvatures, $\begin{matrix}{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\frac{\left( {1 - v_{s}} \right)\alpha_{s}}{{\left( {1 + v_{s}} \right)\alpha_{s}} - {\left( {1 + v_{f}} \right)\alpha_{f}}}\left( {\kappa_{rr} - \kappa_{\theta\theta}} \right)}} & (3.9)\end{matrix}$This relation is independent of the thin film and substrate radii, andis identical to its counterpart for the thin film and substrate with thesame radii subjected to nonuniform temperature change. The sum ofthin-film stresses σ_(rr) ^((f))+σ_(θθ) ^((f)) is related to the sum ofsubstrate curvatures κ_(rr)+κ_(θθ) by $\begin{matrix}{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\begin{Bmatrix}{\kappa_{rr} + \kappa_{\theta\theta} +} \\{{\begin{bmatrix}{\frac{1 - v_{s}}{\quad{1 + v_{s}}} -} \\\frac{\left( {1 - v_{f}} \right)\quad\alpha_{s}}{{\left( {1 + v_{s}} \right)\quad\alpha_{s}} - {\left( {1 + v_{f}} \right)\quad\alpha_{f}}}\end{bmatrix}\left( {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}} \right)} -} \\{\frac{1 + v_{s}}{2}\frac{\left( {1 - v_{f}} \right)\quad\alpha_{s}}{{\left( {1 + v_{s}} \right)\quad\alpha_{s}} - {\left( {1 + v_{f}} \right)\quad\alpha_{f}}}\left( {1 - \frac{R_{f}^{2}}{R_{s}^{2}}} \right)\overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}}\end{Bmatrix}}},} & (3.10)\end{matrix}$where$\overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}} = {\frac{1}{A_{substrate}}\underset{A_{substrate}}{\int\int}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)\eta{\mathbb{d}\eta}{\mathbb{d}\theta}}$is the average substrate curvature. The above equation depends on thethin film and substrate radii, and is different from its counterpart forthe thin film and substrate with the same radii subjected to nonuniformtemperature change.

The interface stress τ(r) can be directly related to substratecurvatures via $\begin{matrix}{\tau = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\frac{\mathbb{d}}{\mathbb{d}r}{\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right).}}} & (3.11)\end{matrix}$This once again provides a remarkably simple way to estimate theinterface shear stress from the radial gradient of the sum of twosubstrate curvatures. Equation (3.10) is independent of the thin filmand substrate radii, and is identical to its counterpart for the thinfilm and substrate with the same radii subjected to temperature change.3.3 Discussion

As described above in this application, unlike Stoney's formula in(1.1), the thin film stresses exhibit nonlocal dependency on thesubstrate curvatures, i.e., stress components at a point on the filmdepend on both the local value of the curvature components (at the samepoint) and on the value of curvatures of all other points on the platesystem (non-local dependence). This demonstrates that analyses methodsbased on Stoney's approach and some of its various extensions are notadequate for handling the non-locality of the stress/curvaturedependence and may result in substantial stress prediction errors ifsuch analyses are applied locally. The presence of non-localcontributions, and the stress dependence on all curvature components,necessitates the use of full-field curvature measurement (over theentire surface of the plate system) in order to determine the filmstresses. Furthermore, the shear stress along the film/substrateinterface is proportional to the radial derivative of the firstcurvature invariant κ_(rr)+κ_(θθ). This provides a means to determinethe interface shear stress that is responsible for the delamination ofthin film/substrate systems.

Based on the analysis in this section, in thin film and substrate ofdifferent radii, for thin film subjected to nonuniform, axisymmetricmisfit strain, the film and substrate radii have no effect on therelation between thin film stresses and substrate curvatures, i.e., thestress-curvature relation is identical to that for thin film andsubstrate of the same radius. This conclusion is particularly useful tothe determination of thin film stresses from substrate curvatures sinceone only needs to know the local and average curvatures of thesubstrate, and not the thin film and substrate radii.

For thin film and substrate subjected to nonuniform, axisymmetrictemperature change, the film and substrate radii still do not affect thedifference in film stresses σ_(rr) ^((f))−σ_(θθ) ^((f)), but theyinfluence the sum of film stresses σ_(rr) ^((f))+σ_(θθ) ^((f)). Thereason for this difference between misfit strain and temperature changeis the former occurs only in the film, while the latter occurs in boththe thin film and substrate, and the nonuniform temperature change inthe substrate leads to stresses and curvatures that depend on the radii.

For both axisymmetric misfit strain and temperature change, theinterface shear stress is proportional to the radial gradient ofcurvature sum, and is independent of the thin film and substrate radii.

4. Techniques for Analysis of Stresses in Thin Film/Substrate SystemsFeaturing Arbitrary Film Thickness and Misfit Strain Distributions

The analysis in this section further relaxes the assumption (i) to studyarbitrarily nonuniform thickness of the thin film. In practice substratesystems, the thickness of a film formed on the substrate can vary due tovarious factors and thus the effects of the film thickness variationneed to take in to account in analyzing the stresses in such a system.As an example, we consider the case of nonuniform film thickness and thethin film/substrate system subject to arbitrary misfit strain field inthe thin film. Examples described in this section relate film stressesand system curvatures to the misfit strain distribution for arbitrarilynonuniform film thickness, and derive a relation between the filmstresses and the system curvatures that would allow for the accurateexperimental inference of film stress from full-field and real-timecurvature measurements.

4.1 Governing Equation

Consider a thin film of nonuniform thickness h_(f)(r,θ) which isdeposited on a circular substrate of constant thickness h_(s) and radiusR, where r and θ are the polar coordinates (FIG. 1). The film is verythin, h_(f)<<h_(s), such that it is modeled as a membrane, and issubject to arbitrary misfit strain distribution ε^(m)(r,θ). Thesubstrate is modeled as a plate since h_(s)<<R. The Young's modulus andPoisson's ratio of the film and substrate are denoted byE_(f),v_(f),E_(s) and v_(s), respectively.

Let u_(r) ^((f)), u_(θ) ^((f)), u_(r) ^((s)) and u_(θ) ^((s)) denote thein-plane displacements in the thin film and substrate along the radial(r) and circumferential (θ) directions, respectively. The in-planemembrane strains are obtained from ε_(αβ)=(u_(α,β)+u_(β,α))/2 forinfinitesimal deformation and rotation, where α,β=r,θ. The linearelastic constitutive model, together with the vanishing out-of-planestress σ_(zz)=0, give the in-plane stresses as${\sigma_{\alpha\beta} = {\frac{E}{1 - v^{2}}\left\lbrack {{\left( {1 - v} \right)ɛ_{\alpha\beta}} + {v\quad ɛ_{\kappa\kappa}\delta_{\alpha\beta}} - {\left( {1 + v} \right)ɛ^{m}\delta_{\alpha\beta}}} \right\rbrack}},$where E, v=E_(f), v_(f) in the thin film and E_(s), v_(s) in thesubstrate, and the misfit strain ε^(m) is only in the thin film. Theaxial forces in the thin film and substrate are $\begin{matrix}{{N_{r} = {\frac{Eh}{1 - v^{2}}\left\lbrack {\frac{\partial u_{r}}{\partial r} + {v\left( {\frac{u_{r}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}}{\partial\theta}}} \right)} - {\left( {1 + v} \right)ɛ^{m}}} \right\rbrack}}{N_{\theta} = {\frac{Eh}{1 - v^{2}}\left\lbrack {\frac{\partial u_{r}}{\partial r} + \frac{u_{r}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}}{\partial\theta}} - {\left( {1 + v} \right)ɛ^{m}}} \right\rbrack}}{{N_{r\quad\theta} = {\frac{Eh}{2\left( {1 + v} \right)}\left( {{\frac{1}{r}\frac{\partial u_{r}}{\partial\theta}} + \frac{\partial u_{\theta}}{\partial r} - \frac{u_{\theta}}{r}} \right)}},}} & (2.1)\end{matrix}$where h=h_(f) in the thin film and h_(s) in the substrate, and onceagain the misfit strain ε^(m) is only in the thin film.

Let w denote the lateral displacement in the normal (z) direction. Thecurvatures are given by κ_(αβ)=w,_(αβ). The bending moments in thesubstrates are $\begin{matrix}{{M_{r} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\left\lbrack {\frac{\partial^{2}w}{\partial r^{2}} + {v_{s}\left( {{\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}} \right\rbrack}}{M_{\theta} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\left( {{v_{s}\frac{\partial^{2}w}{\partial r^{2}}} + {\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}}} \right)}}{M_{r\quad\theta} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 + v_{s}} \right)}\frac{\partial}{\partial r}{\left( {\frac{1}{r}\frac{\partial w}{\partial\theta}} \right).}}}} & (2.2)\end{matrix}$

For non-uniform misfit strain distribution ε^(m)=ε^(m)(r,θ), the shearstresses at the film/substrate interface do not vanish, and are denotedby τ_(r) and τ_(θ). The in-plane force equilibrium equations for thethin film and substrate, accounting for the effect of interface shearstresses τ_(r) and τ_(θ), become $\begin{matrix}{{{\frac{\partial N_{r}}{\partial r} + \frac{N_{r} - N_{\theta}}{r} + {{\frac{1}{r}\frac{\partial N_{r\quad\theta}}{\partial\theta}} \mp \tau_{r}}} = 0}{{{\frac{\partial N_{r\quad\theta}}{\partial r} + {\frac{2}{r}N_{r\quad\theta}} + {{\frac{1}{r}\frac{\partial N_{\theta}}{\partial\theta}} \mp \tau_{\theta}}} = 0},}} & (2.3)\end{matrix}$where the minus sign in front of the interface shear stresses is for thethin film, and the plus sign is for the substrate. The moment andout-of-plane force equilibrium equations for the substrate are$\begin{matrix}{{{\frac{\partial M_{r}}{\partial r} + \frac{M_{r} - M_{\theta}}{r} + {\frac{1}{r}\frac{\partial M_{r\quad\theta}}{\partial\theta}} + Q_{r} - {\frac{h_{s}}{2}\tau_{r}}} = 0},{{\frac{\partial M_{r\quad\theta}}{\partial r} + {\frac{2}{r}M_{r\quad\theta}} + {\frac{1}{r}\frac{\partial M_{\theta}}{\partial\theta}} + Q_{\theta} - {\frac{h_{s}}{2}\tau_{\theta}}} = 0},} & (2.4) \\{{{\frac{\partial Q_{r}}{\partial r} + \frac{Q_{r}}{r} + {\frac{1}{r}\frac{\partial Q_{\theta}}{\partial\theta}}} = 0},} & (2.5)\end{matrix}$where Q_(r) and Q_(θ) are the shear forces normal to the neutral axis.

The substitution of Eq. (2.1) into (2.3) yields the governing equationsfor u_(r), u_(θ), τ_(r), and τ_(θ) $\begin{matrix}{{{{\frac{\partial}{\partial r}\left\{ {h_{f}\left\lbrack {\frac{\partial u_{r}^{(f)}}{\partial r} + \frac{u_{r}^{(f)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(f)}}{\partial\theta}}} \right\rbrack} \right\}} + {\frac{1 - v_{f}}{2}\frac{h_{f}}{r^{2}}\left\{ {\frac{\partial^{2}u_{r}^{(f)}}{\partial\theta^{2}} - {\frac{\partial}{\partial r}\left\lbrack {r\frac{\partial u_{\theta}^{(f)}}{\partial\theta}} \right\rbrack}} \right\}} + {\frac{1 - v_{f}}{2}\left\langle {{\frac{\partial h_{f}}{\partial\theta}\left\{ {{\frac{\partial}{\partial r}\left\lbrack \frac{u_{\theta}^{(f)}}{r} \right\rbrack} + {\frac{1}{r^{2}}\frac{\partial u_{r}^{(f)}}{\partial\theta}}} \right\}} - {\frac{2}{r}{\frac{\partial h_{f}}{\partial r}\left\lbrack {u_{r}^{(f)} + \frac{\partial u_{\theta}^{(f)}}{\partial\theta}} \right\rbrack}}} \right\rangle}} = {{\frac{1 - v_{f}^{2}}{E_{f}}\tau_{r}} + {\left( {1 + v_{f}} \right)\frac{\partial\left( {h_{f}ɛ^{m}} \right)}{\partial r}}}},} & \left( {2.6a} \right) \\{{{{\frac{1}{r}\frac{\partial}{\partial\theta}\left\{ {h_{f}\left\lbrack {\frac{\partial u_{r}^{(f)}}{\partial r} + \frac{u_{r}^{(f)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(f)}}{\partial\theta}}} \right\rbrack} \right\}} + {\frac{1 - v_{f}}{2}h_{f}\left\langle {{- {\frac{\partial}{\partial r}\left\lbrack {\frac{1}{r}\frac{\partial u_{r}^{(f)}}{\partial\theta}} \right\rbrack}} + {\frac{\partial}{\partial r}\left\{ {\frac{1}{r}{\frac{\partial}{\partial r}\left\lbrack {ru}_{\theta}^{(f)} \right\rbrack}} \right\}}} \right\rangle} + {\frac{1 - v_{f}}{2}\left\langle {{\frac{\partial h_{f}}{\partial r}\left\{ {{\frac{1}{r}\frac{\partial u_{r}^{(f)}}{\partial\theta}} + {r{\frac{\partial}{\partial r}\left\lbrack \frac{u_{\theta}^{(f)}}{r} \right\rbrack}}} \right\}} - {\frac{2}{r}\frac{\partial h_{f}}{\partial\theta}\frac{\partial u_{r}^{(f)}}{\partial r}}} \right\rangle}} = {{\frac{1 - v_{f}^{2}}{E_{f}}\tau_{\theta}} + {\left( {1 + v_{f}} \right)\frac{1}{r}\frac{\partial\left( {h_{f}ɛ^{m}} \right)}{\partial\theta}}}},} & \left( {2.6b} \right) \\{{{{\frac{\partial}{\partial r}\left\lbrack {\frac{\partial u_{r}^{(s)}}{\partial r} + \frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right\rbrack} + {\frac{1 - v_{s}}{2}\frac{1}{r^{2}}\left\{ {\frac{\quad{\partial^{2}u_{\quad r}^{(s)}}}{\partial\theta^{2}} - {\frac{\partial}{\partial r}\left\lbrack {r\frac{\partial u_{\theta}^{(s)}}{\partial\theta}} \right\rbrack}} \right\}}} = {{- \frac{1 - v_{s}^{2}}{E_{s}h_{s}}}\tau_{r}}},} & \left( {2.7a} \right) \\{{{\frac{1}{r}{\frac{\partial}{\partial\theta}\left\lbrack {\frac{\partial u_{r}^{(s)}}{\partial r} + \frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right\rbrack}} + {\frac{1 - v_{s}}{2}\left\langle {{- {\frac{\partial}{\partial r}\left\lbrack {\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial\theta}} \right\rbrack}} + {\frac{\partial}{\partial r}\left\{ {\frac{1}{r}{\frac{\partial}{\partial r}\left\lbrack {ru}_{\theta}^{(s)} \right\rbrack}} \right\}}} \right\rangle}} = {{- \frac{1 - v_{s}^{2}}{E_{s}h_{s}}}{\tau_{\theta}.}}} & \left( {2.7b} \right)\end{matrix}$

Elimination of Q_(r) and Q_(θ) from Eqs. (2.4) and (2.5), together withEq. (2.2), gives the governing equation for w, τ_(r) and τ_(r):$\begin{matrix}{{{{\nabla^{2}\left( {\nabla^{2}w} \right)} = {\frac{6\left( {1 - v_{s}^{2}} \right)}{E_{s}h_{s}^{2}}\left( {\frac{\partial\tau_{r}}{\partial r} + \frac{\tau_{r}}{r} + {\frac{1}{r}\frac{\partial\tau_{\theta}}{\partial\theta}}} \right)}},{where}}{\nabla^{2}{= {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}{\frac{\partial^{2}}{\partial\theta^{2}}.}}}}}} & (2.8)\end{matrix}$

The continuity of displacements across the film/substrate interfacerequires $\begin{matrix}{{u_{r}^{(f)} = {u_{r}^{(s)} - {\frac{h_{s}}{2}\frac{\partial w}{\partial r}}}},{u_{\theta}^{(f)} = {u_{\theta}^{(s)} - {\frac{h_{s}}{2}\frac{1}{r}{\frac{\partial w}{\partial\theta}.}}}}} & (2.9)\end{matrix}$

Equations (2.6)-(2.9) constitute seven ordinary 15 differentialequations for seven variables, namely u_(r) ^((f)), u_(θ) ^((f)), u_(r)^((s)), u_(θ) ^((s)), w, τ_(r) and τ_(θ). For the limit h_(f)/h_(s)<<1,these equations are decoupled such that we can solve u_(r) ^((s)), u_(θ)^((s)) first, then w, followed by u_(r) ^((f)) and u_(θ) ^((f)), andfinally τ_(r) and τ_(θ).

(i) Elimination of τ_(r) and τ_(θ) from (2.6) and (2.7) for thesubstrate yields two equations for u_(r) ^((f)), u_(θ) ^((f)), u_(r)^((s)), and u_(θ) ^((s)). For h_(f)/h_(s)<<1, u_(r) ^((f)) and u_(θ)^((f)) disappear in these two equations, which give the governingequations for u_(r) ^((s)) and u_(θ) ^((s)) $\begin{matrix}{{{{\frac{\partial}{\partial r}\left\lbrack {\frac{\partial u_{r}^{(s)}}{\partial r} + \frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right\rbrack} + {\frac{1 - v_{s}}{2}\frac{1}{r^{2}}\left\{ {\frac{\partial^{2}u_{r}^{(s)}}{\partial\theta^{2}} - {\frac{\partial}{\partial r}\left\lbrack {r\frac{\partial u_{\theta}^{(s)}}{\partial\theta}} \right\rbrack}} \right\}}} = {\frac{E_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{\partial}{\partial r}\left( {h_{f}ɛ^{m}} \right)}},} & \left( {2.10a} \right) \\{{{\frac{1}{r}{\frac{\partial}{\partial\theta}\left\lbrack {\frac{\partial u_{r}^{(s)}}{\partial r} + \frac{u_{r}^{(s)}}{r} + {\frac{1}{r}\frac{\partial u_{\theta}^{(s)}}{\partial\theta}}} \right\rbrack}} + {\frac{1 - v_{s}}{2}\left\langle {{- {\frac{\partial}{\partial r}\left\lbrack {\frac{1}{r}\frac{\partial u_{r}^{(s)}}{\partial\theta}} \right\rbrack}} + {\frac{\partial}{\partial\theta}\left\{ {\frac{1}{r}{\frac{\partial}{\partial r}\left\lbrack {ru}_{\theta}^{(s)} \right\rbrack}} \right\}}} \right\rangle}} = {\frac{E_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}\frac{1}{r}\frac{\partial}{\partial\theta}{\left( {h_{f}ɛ^{m}} \right).}}} & \left( {2.10b} \right)\end{matrix}$

(ii) Elimination of u_(r) ^((f)) and u_(θ) ^((f)) from (2.6) and (2.9)gives τ_(r) and τ_(θ) in terms of u_(r) ^((s)), u_(θ) ^((s)) and w (andε^(m)). Substitution of τ_(r) and τ_(θ) into (2.8) yields the followinggoverning equation for w $\begin{matrix}{{\nabla^{2}\left( {\nabla^{2}w} \right)} = {{- 6}\frac{E_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}{{\nabla^{2}\left( {h_{f}ɛ^{m}} \right)}.}}} & (2.11)\end{matrix}$

(iii) The continuity condition (2.9) gives u_(r) ^((f)) and u_(θ)^((f)). The leading terms of the interface shear stresses τ_(r) andτ_(θ) are then obtained from Eq. (2.6) as $\begin{matrix}{{\tau_{r} = {{- \frac{E_{f}}{1 - v_{f}}}\frac{\partial\left( {h_{f}ɛ^{m}} \right)}{\partial r}}},{\tau_{\theta} = {{- \frac{E_{f}}{1 - v_{f}}}\frac{1}{r}{\frac{\partial\left( {h_{f}ɛ^{m}} \right)}{\partial\theta}.}}}} & (2.12)\end{matrix}$

Equations (2.10)-(2.12) show that the film thickness h_(f) alwaysappears together with the misfit strain ε^(m). The interface shearstresses are proportional to the gradients of h_(f)ε^(m), and theyvanish only for uniform misfit strain and uniform film thickness.

The boundary conditions at the free edge r=R require that the net forcesand net moments vanish,N _(r) ^((f)) +N _(r) ^((s))=0 and N _(rθ) ^((f)) +N _(rθ) ^((s))=0,  (2.13)$\begin{matrix}{{M_{r} - {\frac{h_{s}}{2}N_{r}^{(f)}}} = {{{0\quad{and}\quad Q_{r}} - {\frac{1}{r}\frac{\partial}{\partial\theta}\left( {M_{r\quad\theta} - {\frac{h_{s}}{2}N_{r\quad\theta}^{(f)}}} \right)}} = 0.}} & (2.14)\end{matrix}$4.2. Thin-Film Stresses and Substrate Curvatures

Equations (2.10)-(2.12) and boundary conditions (2.13) and (2.14) can besolved in the same way as that for the uniform 15 film thickness butnon-uniform misfit strain (Ngo et al., 2006) by replacing the misfitstrain ε^(m) with h_(f)ε^(m), where h_(f) is the film thickness. Weexpand h_(f)ε^(m) to the Fourier series as $\begin{matrix}{{{h_{f}ɛ^{m}} = {{\sum\limits_{n = 0}^{\infty}{\left( {h_{f}ɛ^{m}} \right)_{c}^{(n)}(r)\quad\cos\quad n\quad\theta}} + {\sum\limits_{n = 1}^{\infty}{\left( {h_{f}ɛ^{m}} \right)_{s}^{(n)}(r)\sin\quad n\quad\theta}}}},{{{where}\quad\left( {h_{f}ɛ^{m}} \right)_{c}^{(0)}(r)} = {\frac{1}{2\quad\pi}{\int_{0}^{2\quad\pi}{h_{f}ɛ^{m}{\mathbb{d}\theta}}}}},{{\left( {h_{f}ɛ^{m}} \right)_{c}^{(n)}(r)} = {{\frac{1}{\pi}{\int_{0}^{2\quad\pi}{h_{f}ɛ^{m}\cos\quad n\quad\theta{\mathbb{d}\theta}\quad\left( {n \geq 1} \right)\quad{{and}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}(r)}}} = {\frac{1}{\pi}{\int_{0}^{2\quad\pi}{h_{f}ɛ^{m}\sin\quad n\quad\theta{\mathbb{d}\theta}\quad{\left( {n \geq 1} \right).}}}}}}} & (3.1)\end{matrix}$

The substrate curvatures${\kappa_{rr} = \frac{\partial^{2}w}{\partial r^{2}}},{\kappa_{\theta\quad\theta} = {{{\frac{1}{r}\frac{\partial w}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}\quad{and}\quad\kappa_{r\quad\theta}}} = {\frac{\partial}{\partial r}\left( {\frac{1}{r}\frac{\partial w}{\partial\theta}} \right)}}}$are related to h_(f)ε^(m) by $\begin{matrix}{{{\kappa_{rr} + \kappa_{\theta\theta}} = {{- 12}\quad\frac{E_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}*\begin{Bmatrix}\begin{matrix}{{h_{f}ɛ^{m}} - {\frac{1 - v_{s}}{2}\left( {{h_{f}ɛ^{m}} - \overset{\_}{h_{f}ɛ^{m}}} \right)} +} \\{\frac{1 - v_{s}^{2}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\left( {n + 1} \right)\frac{r^{n}}{R^{{2\quad n} - 2}}}}}\end{matrix} \\\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}(\eta){\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}(\eta){\mathbb{d}\eta}}}}\end{bmatrix}\end{Bmatrix}}},} & \left( {3.2\quad a} \right) \\{{{\kappa_{rr} - \kappa_{\theta\theta}} = {{- 6}\quad\frac{E_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}*\begin{Bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h_{f}ɛ^{m}} - {\frac{2}{r^{2}}{\int_{0}^{r}{{\eta\left( {h_{f}ɛ^{m}} \right)}_{c}^{(0)}{\mathbb{d}\eta}}}} +} \\{\frac{1 - v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}}\end{matrix} \\{\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix} -}\end{matrix} \\{{\sum\limits_{n = 1}^{\quad\infty}{\frac{n + 1}{\quad r^{n - 2}}\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}} -}\end{matrix} \\{\sum\limits_{n = 1}^{\quad\infty}{\left( {n - 1} \right)\quad{r^{n - 2}\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}}\end{Bmatrix}}},} & \left( {3.2\quad b} \right) \\{{\kappa_{r\quad\theta} = {3\quad\frac{E_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}^{2}}*\begin{Bmatrix}\begin{matrix}\begin{matrix}{\frac{1 - v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}} \\{\begin{bmatrix}{{\sin\quad n\quad\theta{\int_{0}^{R}{{\eta^{n - 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix} +}\end{matrix} \\{{\sum\limits_{n = 1}^{\quad\infty}{\frac{n + 1}{\quad r^{n - 2}}\begin{bmatrix}{{\sin\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{0}^{r}{{\eta^{n - 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}} -}\end{matrix} \\{\sum\limits_{n = 1}^{\quad\infty}{\left( {n - 1} \right)\quad{r^{n - 2}\begin{bmatrix}{{\sin\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}}\end{Bmatrix}}},} & \left( {3.2\quad c} \right)\end{matrix}$where$\overset{\_}{h_{f}ɛ^{m}} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{h_{f}ɛ^{m}{\mathbb{d}A}}}}}$is the average of h_(f)ε^(m) over the entire area A of the thin film,and h_(f)ε^(m) is also related to (h_(f)ε^(m))_(c) ⁽⁰⁾ by$\overset{\_}{h_{f}ɛ^{m}} = {\frac{2}{R^{2}}{\int_{0}^{R}{{\eta\left( {h_{f}ɛ^{m}} \right)}_{c}^{(0)}(\eta){{\mathbb{d}\eta}.}}}}$

The stresses in the thin film are related to h_(f)ε^(m) by$\begin{matrix}{\quad{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{f}}{1 - v_{f}}\left( {{- 2}\quad ɛ^{m}} \right)}},}} & \left( {3.3\quad a} \right) \\{{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {4\quad E_{f}\frac{E_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}*\begin{Bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}{{h_{f}ɛ^{m}} - {\frac{2}{r^{2}}{\int_{0}^{r}{{\eta\left( {h_{f}ɛ^{m}} \right)}_{c}^{(0)}{\mathbb{d}\eta}}}} -} \\{{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}} -}\end{matrix} \\{{\sum\limits_{n = 1}^{\quad\infty}{\left( {n - 1} \right)\quad{r^{n - 2}\begin{bmatrix}{{\cos\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}} -}\end{matrix} \\{\frac{v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}}\end{matrix} \\\begin{bmatrix}{{\cos\quad n\quad\theta\quad{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}\quad ɛ^{m}} \right)}_{c}^{(n)}\quad{\mathbb{d}\eta}}}} +} \\{\sin\quad n\quad\theta\quad{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}\quad ɛ^{m}} \right)}_{s}^{(n)}\quad{\mathbb{d}\eta}}}}\end{bmatrix}\end{Bmatrix}}},} & \left( {3.3\quad b} \right) \\{\sigma_{r\quad\theta}^{(f)} = {2\quad E_{f}\frac{E_{f}}{1 - v_{f}^{2}}\frac{1 - v_{s}^{2}}{E_{s}h_{s}}*\begin{Bmatrix}\begin{matrix}\begin{matrix}{{- {\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{r^{n + 2}}\begin{bmatrix}{{\sin\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{0}^{r}{{\eta^{n + 1}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}} +} \\{{\sum\limits_{n = 1}^{\quad\infty}{\left( {n - 1} \right)\quad{r^{n - 2}\begin{bmatrix}{{\sin\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{c}^{(n)}{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta{\int_{r}^{R}{{\eta^{1 - n}\left( {h_{f}ɛ^{m}} \right)}_{s}^{(n)}{\mathbb{d}\eta}}}}\end{bmatrix}}}} +}\end{matrix} \\{\frac{v_{s}}{3 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\frac{n + 1}{R^{n + 2}}\left\lbrack {{n\quad\frac{r^{n}}{R^{n}}} - {\left( {n - 1} \right)\frac{r^{n - 2}}{R^{n - 2}}}} \right\rbrack}}}\end{matrix} \\\begin{bmatrix}{{\sin\quad n\quad\theta\quad{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}\quad ɛ^{m}} \right)}_{c}^{(n)}\quad{\mathbb{d}\eta}}}} -} \\{\cos\quad n\quad\theta\quad{\int_{0}^{R}{{\eta^{n + 1}\left( {h_{f}\quad ɛ^{m}} \right)}_{s}^{(n)}\quad{\mathbb{d}\eta}}}}\end{bmatrix}\end{Bmatrix}}} & \left( {3.3\quad c} \right)\end{matrix}$

For uniform misfit strain distribution ε^(m)=constant and uniform filmthickness h_(f)=constant, the interface shear stresses in Eq. (2.12)vanish. The curvatures in (3.2) become${\kappa = {\kappa_{rr} = {\kappa_{\theta\theta} = {{- 6}\quad\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s}^{2}}ɛ^{m}}}}},{\kappa_{r\quad\theta} = 0.}$

The stresses in the thin film in (3.3) become${\sigma^{(f)} = {\sigma_{rr}^{(f)} = {\sigma_{\theta\quad\theta}^{(f)} = {\frac{E_{f}}{1 - v_{f}}\left( {- ɛ^{m}} \right)}}}},{\sigma_{r\quad\theta}^{(f)} = 0.}$

For this special case only, both stress and curvature states becomeequi-biaxial. The elimination of misfit strain ε^(m) and film thicknessh_(f) from the above two equations yields a simple relation${\sigma^{(f)} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\kappa}},$which is exactly the Stoney formula in Eq. (1.1), and it has been usedto estimate the thin-film stress σ^((f)) from the substrate curvature κ,if the misfit strain, film thickness, stress and curvature are allconstant and if the plate system shape is spherical. In the following,we extend such a relation for arbitrary non-uniform misfit straindistribution and non-uniform film thickness.4.3 Extension of Stoney Formula for Non-Uniform Misfit StrainDistribution and Non-Uniform Film Thickness

The stresses and curvatures are all given in terms of misfit strain inthe previous section. We extend the Stoney formula for arbitrarynon-uniform misfit strain distribution and non-uniform film thickness inthis section by establishing the direct relation between the thin-filmstresses and substrate curvatures.

We first define the coefficients C_(n) and S_(n) related to thesubstrate curvatures by $\begin{matrix}{{C_{n} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\quad\theta}} \right)\left( \frac{\eta}{R} \right)^{n}\cos\quad n\quad\varphi{\mathbb{d}A}}}}}},{S_{n} = {\frac{1}{\pi\quad R^{2}}{\int{\int_{A}{\left( {\kappa_{rr} + \kappa_{\theta\quad\theta}} \right)\left( \frac{\eta\quad}{R} \right)^{n}\sin\quad n\quad\varphi{\mathbb{d}A}}}}}},} & (4.1)\end{matrix}$where the integration is over the entire area A of the thin film, anddA=ηdηdφ. Since both the substrate curvatures and film stresses dependon the misfit strain ε^(m) and film thickness h_(f), elimination ofh_(f)ε^(m) gives the film stress in terms of substrate curvatures by$\begin{matrix}{{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\begin{Bmatrix}{{4\left( {\kappa_{\quad{rr}} - \kappa_{\quad{\theta\theta}}} \right)} -} \\{\sum\limits_{n\quad = \quad 1}^{\quad\infty}{\left( {n + 1} \right)\left\lbrack {{n\quad\left( \frac{r}{\quad R} \right)^{n}} - {\left( {n - 1} \right)\left( \frac{r}{R} \right)^{n - 2}}} \right\rbrack}} \\\left( {{C_{\quad n}\cos\quad n\quad\theta} + {S_{\quad n}\sin\quad n\quad\theta}} \right)\end{Bmatrix}}},} & \left( {4.2a} \right) \\{{\sigma_{r\quad\theta}^{(f)} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\begin{Bmatrix}{{4\kappa_{\quad{r\quad\theta}}} +} \\{\frac{1}{\quad 2}{\sum\limits_{n\quad = \quad 1}^{\quad\infty}{\left( {n + 1} \right)\left\lbrack {\left( {n\left( \quad\frac{r}{\quad R} \right)} \right)^{n} - {\left( {n - 1} \right)\left( \quad\frac{r}{\quad R} \right)^{n\quad - \quad 2}}} \right\rbrack}}} \\\left( {{C_{\quad n}\sin\quad n\quad\theta} + {S_{\quad n}\cos\quad n\quad\theta}} \right)\end{Bmatrix}}},} & \left( {4.2b} \right) \\{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}h_{s}^{2}}{6{h_{f}\left( {1 - v_{s}} \right)}}\begin{bmatrix}{\kappa_{rr} + \kappa_{\theta\theta} + {\frac{1 - v_{s}}{1 + v_{s}}\left( {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}} \right)} -} \\{\frac{1 - v_{s}}{1 + v_{s}}{\sum\limits_{n = 1}^{\infty}{\left( {n + 1} \right)\left( \frac{r}{R} \right)^{n}\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)}}}\end{bmatrix}}},} & \left( {4.2c} \right)\end{matrix}$where$\overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}} = {C_{0} = {\frac{1}{\pi\quad R^{2}}\underset{A}{\int\int}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right){\mathbb{d}A}}}$is the average curvature over A entire area A of the thin film. Equation(4.2) provides direct relations between individual film stresses andsubstrate curvatures. Stresses at a point in the thin film depend notonly on curvatures at the same point (local dependence), but also on thecurvatures in the entire substrate (non-local dependence) via thecoefficients C_(n) and S_(n). Eq. (4.2b) for shear stress σ_(rθ) ^((f))and Eq. (4.2a) for the difference in normal stresses σ_(r) ^((f))−σ_(θθ)^((f)) are independent of the thin film thickness h_(f), but Eq. (4.2c)for the sum of normal stresses σ_(rr) ^((f))+σ_(θθ) ^((f)) is inverselyproportional to the local film thickness h_(f) at the same point.

The interface shear stresses τ_(r) and τ_(θ) can also be directlyrelated to substrate curvatures via $\begin{matrix}{{\tau_{r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\begin{bmatrix}{{\frac{\partial}{\partial r}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} -} \\{\frac{\left( {1 - v_{s}} \right)}{2\quad R}{\sum\limits_{n = 1}^{\quad\infty}{{n\left( {n + 1} \right)}\left( {{C_{n}\cos\quad n\quad\theta} + {S_{n}\sin\quad n\quad\theta}} \right)\left( \frac{r}{R} \right)^{n - 1}}}}\end{bmatrix}}},} & \left( {4.3a} \right) \\{{\tau_{\theta} = {\frac{E_{s}\quad h_{s}^{2}}{6\quad\left( {1 - v_{s}^{2}} \right)}\begin{bmatrix}{{\frac{1}{r}\frac{\partial}{\partial\theta}\quad\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)} +} \\{\frac{1 - v_{s}}{2\quad R}\quad{\sum\limits_{n = 1}^{\infty}{{n\left( {n + 1} \right)}\quad\left( {{C_{n}\quad\sin\quad n\quad\theta} + {S_{n}\quad\cos\quad n\quad\theta}} \right)\quad\left( \frac{r}{R} \right)^{n - 1}}}}\end{bmatrix}}},} & \left( {4.3b} \right)\end{matrix}$which is also independent of the film thickness h_(f). Equation (4.3)provides a way to determine the interface shear stresses from thegradients of substrate curvatures, and it also displays a non-localdependence via the coefficients C_(n) and S_(n).

Since interfacial shear stresses are responsible for promoting systemfailures through delamination of the thin film from the substrate, Eq.(4.3) has particular significance. It shows that such stresses arerelated to the gradients of κ_(rr)+κ_(θθ) and not to its magnitude asmight have been expected of a local, Stoney-like formulation. Equation(4.3) provides an easy way of inferring these special interfacial shearstresses once the full-field curvature information is available. As aresult, the methodology also provides a way to evaluate the risk of andto mitigate such important forms of failure.

It can be shown that the relations between the film stresses andsubstrate curvatures given in the form of infinite series in (4.2) and(4.3) can be equivalently expressed in the form of integration:$\begin{matrix}{{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\begin{Bmatrix}{{4\left( {\kappa_{\quad{rr}} - \kappa_{\quad{\theta\theta}}} \right)} - {\frac{1}{\quad{\pi\quad R^{2}}}\underset{A}{\int\int}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)}} \\{\frac{\frac{\eta}{R}{F_{minus}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\frac{\eta\quad r}{R^{2}}{\cos\left( {\varphi - \theta} \right)}} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}\end{Bmatrix}}},} & \left( {4.4a} \right) \\{{\sigma_{r\quad\theta}^{(f)} = {{- \frac{E_{f}h_{s}}{6\left( {1 + v_{f}} \right)}}\begin{Bmatrix}{{4\kappa_{\quad{r\quad\theta}}} - {\frac{1}{2}\frac{1}{\quad{\pi\quad R^{2}}}\quad\underset{A}{\int\int}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)}} \\{\frac{\frac{\eta}{R}{F_{shear}\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}}{\left\lbrack {1 - {2\frac{\eta\quad r}{R^{2}}{\cos\left( {\varphi - \theta} \right)}} + \frac{\eta^{2}r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}\end{Bmatrix}}},} & \left( {4.4b} \right) \\{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}h_{s}}{6{h_{f}\left( {1 - v_{s}} \right)}}*\begin{Bmatrix}{\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}} + {\frac{1 - v_{s}}{1 + v_{s}}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}} - \overset{\_}{\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}}} \right)} -} \\{\frac{1 - v_{s}}{1 + v_{s}}\frac{r}{\pi\quad R^{3}}\underset{A}{\int\int}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} \\{\frac{\frac{\eta}{\quad R}\quad{F_{plus}\left( {\frac{r}{R},\frac{\eta}{\quad R},{\varphi - \theta}} \right)}}{\quad\left\lbrack {1 - {2\quad\frac{\eta\quad r}{\quad R^{\quad 2}}\quad{\cos\left( {\varphi - \theta} \right)}}\quad + \frac{\eta^{\quad 2}\quad r^{\quad 2}}{\quad R^{\quad 4}}} \right\rbrack^{2}}{\mathbb{d}A}}\end{Bmatrix}}},} & \left( {4.4c} \right)\end{matrix}$where functions F_(minus), F_(shear) and F_(plus) are given byF _(minus)(r ₁,η₁,φ₁)=r ₁ ²η₁(6+9η₁ ² +r ₁ ²η₁ ⁴)+r ₁(2+9η₁ ²+6r ₁ ²η₁²+6r ₁ ²η₁ ⁴)cos φ₁F _(shear)(r ₁,η₁,φ₁)=r ₁(2+9η₁ ²−6r ₁ ²η₁ ²) sin φ₁−η₁(3+3r ₁ ²η₁ ²−2r₁ ⁴η₁ ²)sin 2φ₁ +r ₁η₁ ² sin 3φ₁,F _(plus)(r ₁,η₁,φ₁)=2(1+2r ₁ ²η₁ ²)cos φ₁ −r ₁η₁(4+r ₁ ²η₁ ²).   (4.5)

The interface shear stresses can also be related to substrate curvaturesvia integrals as $\begin{matrix}{{\tau_{r} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\begin{Bmatrix}{{\frac{\partial}{\partial r}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} -} \\{\frac{1\quad - \quad v_{\quad s}}{\quad{\pi\quad R^{\quad 3}}}\underset{A}{\int\int}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} \\{\frac{\frac{\eta}{R}F_{radial}\quad\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}{\quad\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\quad\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}\quad r^{2}}{\quad R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}\end{Bmatrix}}},} & \left( {4.6a} \right) \\{{\tau_{\theta} = {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}^{2}} \right)}\begin{Bmatrix}{{\frac{1}{r}\frac{\partial}{\partial\theta}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} -} \\{\frac{1 - v_{s}}{\quad{\pi\quad R^{3}}}\underset{A}{\int\int}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} \\{\frac{\frac{\eta}{R}F_{circumferential}\quad\left( {\frac{r}{R},\frac{\eta}{R},{\varphi - \theta}} \right)}{\quad\left\lbrack {1 - {2\quad\frac{\eta\quad r}{R^{2}}\quad\cos\quad\left( {\varphi - \theta} \right)} + \frac{\eta^{2}\quad r^{2}}{R^{4}}} \right\rbrack^{3}}{\mathbb{d}A}}\end{Bmatrix}}},{where}} & \left( {4.6b} \right) \\{{{F_{radial}\left( {r_{1},\eta_{1},\varphi_{1}} \right)} = {{\left( {1 + {3r_{1}^{2}\eta_{1}^{2}}} \right)\cos\quad\varphi_{1}} - {r_{1}{\eta_{1}\left( {3 + {r_{1}^{2}\eta_{1}^{2}\cos\quad 2\varphi_{1}}} \right)}}}},{{F_{circumferential}\left( {r_{1},\eta_{1},\varphi_{1}} \right)} = {{\left( {1 - {3r_{1}^{2}\eta_{1}^{2}}} \right)\sin\quad\varphi_{1}} + {r_{1}^{3}\eta_{1}^{3}\sin\quad 2\quad{\varphi_{1}.}}}}} & (4.7)\end{matrix}$4.4 Discussion

The dependence of film stresses on substrate curvatures is non-local,i.e., the stress components at a point on the film depend on both thecurvature components at the same point and on the curvatures of allother points on the plate system. The presence of non-localcontributions in such relations also has implications regarding thenature of diagnostic methods needed to perform wafer-level film stressmeasurements. Notably the existence of non-local terms necessitates theuse of full-field methods capable of measuring curvature components overthe entire surface of the plate system (or wafer). Furthermoremeasurement of all independent components of the curvature field isnecessary because the stress state at a point depends on curvaturecontributions (from κ_(rr), κ_(θθ) and κ_(rθ)) from the entire platesurface.

The non-uniformities also result in the shear stresses along the thinfilm/substrate interface. Such interface shear stresses vanish for thespecial case of uniform κ_(rr)+κ_(θθ) in the Stoney formula and itsvarious extensions. Since film delamination is a commonly encounteredform of failure during wafer manufacturing, the ability to estimate thelevel and distribution of such stresses from wafer-level metrology canbe invaluable in enhancing the reliability of such systems.

The present analysis provides a simple way to account for the effect ofnon-uniform film thickness on the Stoney formula. For arbitrarilynon-uniform film thickness, the stress-curvature relations are identicalto their counterparts for uniform film thickness except that thicknessis replaced by its local value. For example, the sum of normal stressesσ_(rr) ^((f))+σ_(θθ) ^((f)) at a point on the film is inverselyproportional to the local film thickness at the same point.

5. Analysis of stresses in a Thin Film/Substrate System With Non-UniformSubstrate Thickness

In practical substrate systems, the thickness of the substrate may varyacross the substrate. The analysis of this section is to relax theremaining portion in assumption (i) (i.e., the assumption of the uniformthickness of the substrate) to account for variations in the thicknessof the substrate. As an example, we consider the case of thinfilm/substrate system with non-uniform substrate thickness subject tonon-uniform misfit strain field in the thin film. Examples described inthis section relate film stresses and system curvatures to the misfitstrain distribution, and derive a relation between the film stresses andthe system curvatures that would allow for the accurate experimentalinference of film stress from full-field and real-time curvaturemeasurements.

5.1 Governing Equations and Boundary Conditions

Consider a thin film of uniform thickness h_(f) which is deposited on acircular substrate of thickness h_(s) and radius R. The substratethickness is non-uniform, but is assumed to be axisymmetrich_(s)=h_(s)(r) for simplicity, where r and θ are the polar coordinates.The film is very thin, h_(f)<<h_(s), such that it is modeled as amembrane, and is subject to non-uniform misfit strain ε_(m). Here themisfit strain is also assumed to be axisymmetric ε_(m)=ε_(m)(r) forsimplicity. The substrate is modeled as a plate since h_(s)<<R. TheYoung's modulus and Poisson's ratio of the film and substrate aredenoted by E_(f),v_(f),E_(s) and v_(s), respectively.

Let u_(f) and u_(s) denote the displacements in the radial direction inthe thin film and substrate, respectively. The in-plane membrane strainsare obtained from ε_(αβ)=(u_(α,β)+u_(β,α))/2 for infinitesimaldeformation and rotation, where α,β=r,θ. The linear elastic constitutivemodel, together with the vanishing out-of-plane stress σ_(zz)=0, givethe in-plane stresses as${\sigma_{\alpha\beta} = {\frac{E}{1 - v^{2}}\left\lbrack {{\left( {1 - v} \right)ɛ_{\alpha\beta}} + {v\quad ɛ_{\kappa\kappa}\delta_{\alpha\beta}} - {\left( {1 + v} \right)ɛ_{m}\delta_{\alpha\beta}}} \right\rbrack}},$where E,v=E_(f), v_(f) in the thin film and E_(s), v_(s) in thesubstrate, and the misfit strain ε^(m) is only in the thin film. Thenon-vanishing axial forces in the thin film and substrate are$\begin{matrix}{{N_{r} = {\frac{Eh}{1 - v^{2}}\left\lbrack {\frac{\mathbb{d}u_{r}}{\mathbb{d}r} + {v\quad\frac{u_{r}}{r}} - {\left( {1 + v} \right)ɛ_{m}}} \right\rbrack}},{N_{\theta} = {\frac{Eh}{1 - v^{2}}\left\lbrack {{v\frac{\mathbb{d}u_{r}}{\mathbb{d}r}} + \frac{u_{r}}{r} - {\left( {1 + v} \right)ɛ_{m}}} \right\rbrack}},} & (2.1)\end{matrix}$where h=h_(f) in the thin film and h_(s)(r) in the substrate, and onceagain the misfit strain ε_(m) is only in the thin film.

Let w denote the lateral displacement in the normal (z) direction. Thecurvatures are given by κ_(αβ)=w,_(αβ). The bending moments in thesubstrates are $\begin{matrix}{{M_{r} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}\left( {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} + {v_{s}\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right)}},{M_{\theta} = {\frac{E_{s}h_{s}^{3}}{12\left( {1 - v_{s}^{2}} \right)}{\left( {{v_{s}\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}} + {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right).}}}} & (2.2)\end{matrix}$

For non-uniform misfit strain distribution ε_(m)=ε_(m)(r), the shearstress along the radial direction at the film/substrate interface doesnot vanish, and is denoted by τ. The in-plane force equilibriumequations for the thin film and substrate, accounting for the effect ofinterface shear stress τ, becomes $\begin{matrix}{{{\frac{\mathbb{d}N_{r}}{\mathbb{d}r} + {\frac{N_{r} - N_{\theta}}{r} \mp \tau}} = 0},} & (2.3)\end{matrix}$where the minus sign in front of the interface shear stress is for thethin film, and the plus sign is for the substrate. The moment andout-of-plane force equilibrium equations for the substrate are$\begin{matrix}{{{\frac{\mathbb{d}M_{r}}{\mathbb{d}r} + \frac{M_{r} - M_{\theta}}{r} + Q - {\frac{h_{s}}{2}\tau}} = 0},} & (2.4) \\{{{\frac{\mathbb{d}Q}{\mathbb{d}r} + \frac{Q}{r}} = 0},} & (2.5)\end{matrix}$where Q is the shear force normal to the neutral axis. Equation (2.5),together with the requirement of finite Q at r=0, gives Q=0.

The substitution of Eq. (2.1) into (2.3) yields the governing equationsfor u and τ $\begin{matrix}{{{\frac{\mathbb{d}^{2}u_{f}}{\mathbb{d}r^{2}} + {\frac{1}{r}\frac{\mathbb{d}u_{f}}{\mathbb{d}r}} - \frac{u_{f}}{r^{2}}} = {{\frac{1 - v_{f}^{2}}{E_{f}h_{f}}\tau} + {\left( {1 + v_{f}} \right)\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}}}},} & (2.6) \\{{{\frac{\mathbb{d}}{\mathbb{d}r}\left\lbrack {h_{s}\left( {\frac{\mathbb{d}u_{s}}{\mathbb{d}r} + \frac{u_{s}}{r}} \right)} \right\rbrack} - {\left( {1 - v_{s}} \right)\frac{\mathbb{d}h_{s}}{\mathbb{d}r}\frac{u_{s}}{r}}} = {\frac{1 - v_{s}^{2}}{E_{s}}{\tau.}}} & (2.7)\end{matrix}$Equations (2.2), (2.4) and (2.5) give the governing equation for w and τ$\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}r}\left\lbrack {h_{s}^{3}\left( {\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}} + {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}} \right)} \right\rbrack} - {\left( {1 - v_{s}} \right)\frac{1}{r}\frac{\mathbb{d}h_{s}^{3}}{\mathbb{d}r}\frac{\mathbb{d}w}{\mathbb{d}r}}} = {\frac{6\left( {1 - v_{s}^{2}} \right)}{E_{s}}h_{s}{\tau.}}} & (2.8)\end{matrix}$

The continuity of displacements across the film/substrate interfacerequires $\begin{matrix}{u_{f} = {u_{s} - {\frac{h_{s}}{2}{\frac{\mathbb{d}w}{\mathbb{d}r}.}}}} & (2.9)\end{matrix}$Equations (2.6)-(2.9) constitute four ordinary differential equations(ODEs) for u_(f), u_(s), w and τ. The ODEs are linear, but havenon-constant coefficients.

The boundary conditions at the free edge r=R require that the net forcesand net moments vanish, $\begin{matrix}{{{N_{r}^{(f)} + N_{r}^{(s)}} = 0},} & (2.10) \\{{{M_{r} - {\frac{h_{s}}{2}N_{r}^{(f)}}} = 0},} & (2.11)\end{matrix}$where the superscripts f and s denote the film and substrate,respectively.5.2 Perturbation Method for Small Variation of Substrate Thickness

In the following we assume small variation of substrate thicknessh _(s) =h _(s0) +Δh _(s) =h _(s0) +βh _(s1),   (3.1)where h_(s0) (=constant) is the average substrate thickness, andΔh_(s)(r) is the substrate thickness variation which satisfies|Δh_(s)|<<h_(s0); Δh_(s)(r) is also written as βh_(s1) in (3.1), where0<β<<1 is a small, positive constant, and h_(s1)=h_(s1)(r) is on thesame order as h_(s0).

We use the perturbation method to solve the ODEs analytically for β<<1.Two possible scenarios are considered separately in the following.

The substrate thickness variation Δh_(s) is on the same order as thethin film thickness h_(f), i.e., Δh_(s)˜h_(f). This is represented by$\beta = {\frac{h_{f}}{h_{s\quad 0}}{\left( {\operatorname{<<}1} \right).}}$For this case the film stresses and system curvatures are identical totheir counterparts for a constant substrate thickness h_(s0). This isbecause the Stoney formula (1.1), as well as all its extensions, holdsonly for thin films, h_(f)<<h_(s). As compared to unity (one), termsthat are on the order of $O\left( \frac{h_{f}}{h_{s}} \right)$are always neglected. In this case the difference between the filmstresses (or system curvatures, . . . ) for non-uniform substratethickness h_(s) and those for constant thickness h_(s0) is on the orderof $O\left( \frac{\Delta\quad h_{s}}{h_{s\quad 0}} \right)$(as compared to unity), which is the same as$O\left( \frac{h_{f}}{h_{s}} \right)$since Δh_(s)˜h_(f), and is therefore negligible.

The substrate thickness variation Δh_(s) is much larger than the thinfilm thickness h_(f), i.e., |Δh|>>h_(f). This is represented by$\begin{matrix}{\frac{h_{f}}{h_{s\quad 0}}{\operatorname{<<}\beta}} & {\left( {\operatorname{<<}1} \right).}\end{matrix}$In the following we focus on this case and use the perturbation method(for β<<1) to obtain the analytical solution.

Elimination of τ from (2.6) and (2.7) yields an equation for u_(f) andu_(s). For h_(f)/h_(s0)<<1, u_(f) disappears in this equation, whichbecomes the governing equation for u_(s) $\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}r}\left\lbrack {h_{s}\left( {\frac{\mathbb{d}u_{s}}{\mathbb{d}r} + \frac{u_{s}}{r}} \right)} \right\rbrack} - {\left( {1 - v_{s}} \right)\frac{\mathbb{d}h_{s}}{\mathbb{d}r}\frac{u_{s}}{r}}} = {\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}}{\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}.}}} & (3.2)\end{matrix}$

The above equation, together with (2.7), gives the interface shearstress $\begin{matrix}{\tau = {{- \frac{E_{f}h_{f}}{1 - v_{f}}}{\frac{\mathbb{d}ɛ_{m}}{\mathbb{d}r}.}}} & (3.3)\end{matrix}$

This is a remarkable result that holds regardless of the substratethickness and boundary conditions at the edge r=R. Therefore, theinterface shear stress is proportional to the gradient of misfit strain.For uniform misfit strain ε_(m)(r)=constant, the interface shear stressvanishes (even for non-uniform substrate thickness).

We use the perturbation method to write u_(s) asu _(s) =u _(s0) +βu _(s1)   (3.4)where β<<1, u_(s0) is the solution for a constant substrate thicknessh_(s0), and is given by (Huang et al., 2005) $\begin{matrix}{{u_{s\quad 0} = {\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}}\left\lbrack {{\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}r}} \right\rbrack}}},} & (3.5)\end{matrix}$and$\overset{\_}{ɛ_{m}} = {\frac{2}{\pi\quad R^{2}}{\int_{0}^{R}{{\eta ɛ}_{m}\quad{\mathbb{d}\eta}}}}$is the average misfit strain in the thin film; u_(s1) in (3.4) is on thesame order as u_(s0). In the following we use u′ to denote$\frac{\mathbb{d}u}{\mathbb{d}r}.$The substitution of (3.4) and (3.5) into (3.2) and the neglect of O(β²)terms give the following linear ODE with constant coefficients foru_(s1) $\begin{matrix}{\left( {u_{s\quad 1}^{\prime} + \frac{u_{s\quad 1}}{r}} \right)^{\prime} = {{\left( {1 - v_{s}} \right)\frac{h_{s\quad 1}^{\prime}}{h_{s\quad 0}}\frac{u_{s\quad 0}}{r}} - {\left\lbrack {\frac{h_{s\quad 1}}{h_{s\quad 0}}\left( {u_{s\quad 0}^{\prime} + \frac{u_{s\quad 0}}{r}} \right)} \right\rbrack^{\prime}.}}} & (3.6)\end{matrix}$

Its general solution is $\begin{matrix}{{{u_{s\quad 1}(r)} = {{{- \frac{h_{s\quad 1}}{h_{s\quad 0}}}u_{s\quad 0}} + {\frac{1}{2r}{\int_{0}^{r}{{\eta\left\lbrack {1 + v_{s} + {\left( {1 - v_{s}} \right)\frac{r^{2}}{\eta^{2}}}} \right\rbrack}\frac{h_{s\quad 1}^{\prime}(\eta)}{h_{s\quad 0}}{u_{s\quad 0}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{A}{2}r}}},} & (3.7)\end{matrix}$where the constant A is to be determined. The total substratedisplacement is then given by u $\begin{matrix}{{u_{s}(r)} = {{\left( {2 - \frac{h_{s}}{h_{s\quad 0}}} \right)u_{s\quad 0}} + {\frac{1}{2r}{\int_{0}^{r}{{\eta\left\lbrack {1 + v_{s} + {\left( {1 - v_{s}} \right)\frac{r^{2}}{\eta^{2}}}} \right\rbrack}\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{u_{s\quad 0}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{\beta\quad A}{2}{r.}}}} & (3.8)\end{matrix}$

The substitution of (3.3) into (2.8) yields the governing equation forthe displacement w′, $\begin{matrix}{{\left\lbrack {h_{s}^{3}\left( {w^{''} + \frac{w^{\prime}}{r}} \right)} \right\rbrack^{\prime} - {\left( {1 - v_{s}} \right)\left( h_{s}^{3} \right)^{\prime}\frac{w^{\prime}}{r}}} = {{- \frac{6E_{f}h_{f}}{1 - v_{f}}}\frac{1 - v_{s}^{2}}{E_{s}}h_{s}{ɛ_{m}^{\prime}.}}} & (3.9)\end{matrix}$

Its perturbation solution can be written asw′=w ₀ ′+βw ₁′,   (3.10)where w₀′ is the solution for a constant substrate thickness h_(s0), andis given by (Huang et al., 2005) $\begin{matrix}{{w_{0}^{\prime} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}{\frac{1 - v_{s}^{2}}{E_{s}h_{0}^{2}}\left\lbrack {{\frac{1}{r}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}} + {\frac{1 - v_{s}}{1 + v_{s}}\frac{\overset{\_}{ɛ_{m}}}{2}r}} \right\rbrack}}},} & (3.11)\end{matrix}$and once again$\overset{\_}{ɛ_{m}} = {\frac{2}{\pi\quad R^{2}}{\int_{0}^{R}{{\eta ɛ}_{m}\quad{\mathbb{d}\eta}}}}$is the average misfit strain in the thin film; w₁′ in (3.10) is on thesame order as w₀′. Equations (3.9)-(3.11) give the following linear ODEwith constant coefficients for w₁′ $\begin{matrix}\begin{matrix}{\left( {w_{1}^{''} + \frac{w_{1}^{\prime}}{r}} \right) = {{{- \frac{6E_{f}h_{f}}{1 - v_{f}}}\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}^{2}}\frac{h_{s\quad 1}}{h_{s\quad 0}}ɛ_{m}^{\prime}} -}} \\{{3\left\lbrack {\frac{h_{s\quad 1}}{h_{s\quad 0}}\left( {w_{0}^{''} + \frac{w_{0}^{\prime}}{r}} \right)} \right\rbrack}^{\prime} +} \\{3\left( {1 - v_{s}} \right)\frac{h_{s\quad 1}^{\prime}}{h_{s\quad 0}}{\frac{w_{0}^{\prime}}{r}.}}\end{matrix} & (3.12)\end{matrix}$

Its general solution is $\begin{matrix}\begin{matrix}{w_{1}^{\prime} = {{{- 3}\frac{h_{s\quad 1}}{h_{s\quad 0}}w_{0}^{\prime}} + {\frac{3}{2r}{\int_{0}^{r}{\eta\left\lbrack {1 + v_{s} + {\left( {1 - v_{s}} \right)\frac{r^{2}}{\eta^{2}}}} \right\rbrack}}}}} \\{{\frac{h_{s\quad 1}^{\prime}(\eta)}{h_{s\quad 0}}{w_{0}^{\prime}(\eta)}\quad{\mathbb{d}\eta}} + {\frac{B}{2}r} +} \\{{3\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}^{2}}\frac{1}{r}{\int_{0}^{r}{{\frac{\mathbb{d}}{\mathbb{d}\eta}\begin{bmatrix}\left( {r^{2} - \eta^{2}} \right) \\\frac{h_{s\quad 1}(\eta)}{h_{s\quad 0}}\end{bmatrix}}{ɛ_{m}(\eta)}\quad{\mathbb{d}\eta}}}},}\end{matrix} & (3.13)\end{matrix}$where the constant B is to be determined. The complete solution for w′is obtained from (3.10) as $\begin{matrix}\begin{matrix}{w_{1}^{\prime} = {{\left( {4 - {3\frac{h_{s}}{h_{s\quad 0}}}} \right)w_{0}^{\prime}} + {\frac{3}{2r}{\int_{0}^{r}{\eta\left\lbrack {1 + v_{s} + {\left( {1 - v_{s}} \right)\frac{r^{2}}{\eta^{2}}}} \right\rbrack}}}}} \\{{\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{w_{0}^{\prime}(\eta)}\quad{\mathbb{d}\eta}} + {\frac{\beta\quad B}{2}r} +} \\{3\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}^{2}}\frac{1}{r}{\int_{0}^{r}{\frac{\mathbb{d}}{\mathbb{d}\eta}\begin{Bmatrix}\left( {r^{2} - \eta^{2}} \right) \\\left\lbrack {\frac{h_{s}(\eta)}{h_{s\quad 0}} - 1} \right\rbrack\end{Bmatrix}{ɛ_{m}(\eta)}\quad{{\mathbb{d}\eta}.}}}}\end{matrix} & (3.14)\end{matrix}$The displacement u_(f) in the thin film is then obtained from u_(s) in(3.8) and w′ in (3.14) via (2.9).

The constants A and B, or equivalently, βA and βB, are determined fromthe boundary conditions (2.10) and (2.11) as $\begin{matrix}{{{\beta\quad A} = {{- \frac{1 - v_{s}}{R^{2}}}{\int_{0}^{R}{\frac{R^{2} - \eta^{2}}{\eta}\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{u_{s\quad 0}(\eta)}\quad{\mathbb{d}\eta}}}}},} & (3.15) \\\begin{matrix}{{\beta\quad B} = {{{- \frac{3\left( {1 - v_{s}} \right)}{R^{2}}}{\int_{0}^{R}{\frac{R^{2} - \eta^{2}}{\eta}\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{w_{0}^{\prime}(\eta)}{\mathbb{d}\eta}}}} -}} \\{6\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s\quad 0}^{2}}\frac{1}{R^{2}}{\int_{0}^{R}{\frac{\mathbb{d}}{\mathbb{d}\eta}\begin{Bmatrix}\begin{bmatrix}{{\left( {1 + v_{s}} \right)R^{2}} +} \\{\left( {1 - v_{s}} \right)\eta^{2}}\end{bmatrix} \\\left\lbrack {\frac{h_{s}(\eta)}{h_{s\quad 0}} - 1} \right\rbrack\end{Bmatrix}}}} \\{{ɛ_{m}(\eta)}\quad{{\mathbb{d}\eta}.}}\end{matrix} & (3.16)\end{matrix}$5.3 Thin-Film Stresses and System Curvatures

The system curvatures$\kappa_{rr} = {{\frac{\mathbb{d}^{2}w}{\mathbb{d}r^{2}}\quad{and}\quad\kappa_{\theta\theta}} = {\frac{1}{r}\frac{\mathbb{d}w}{\mathbb{d}r}}}$are obtained from (3.14). Their sum κ_(Σ)≡κ_(rr)+κ_(θθ) is given interms of the misfit strain by $\begin{matrix}{{\kappa_{\Sigma} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}^{2}}\begin{Bmatrix}{{\left( {3 - {2\frac{h_{s}}{h_{s\quad 0}}}} \right)ɛ_{m}} + {\left\lbrack {4 - {3\frac{h_{s}}{h_{s\quad 0}}} + {\frac{3\left( {1 - v_{s}} \right)}{2}\frac{h_{s} - {h_{s}(0)}}{h_{s\quad 0}}}} \right\rbrack\frac{1 - v_{s}}{1 + v_{s}}\overset{\_}{ɛ_{m}}} +} \\{\int_{0}^{r}{\left\lbrack {{\frac{3\left( {1 - v_{s}} \right)}{\eta^{2}}{\int_{0}^{\eta}{{{\rho ɛ}_{m}(\rho)}{\mathbb{d}\rho}}}} - {ɛ_{m}(\eta)}} \right\rbrack\quad\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{\mathbb{d}\eta}}}\end{Bmatrix}}},} & (4.1)\end{matrix}$where$\overset{\_}{ɛ_{m}} = {\frac{2}{\pi\quad R^{2}}{\int_{0}^{R}{{\eta ɛ}_{m}\quad{\mathbb{d}\eta}}}}$is the average misfit strain in the thin film. The difference of systemcurvatures κ_(Δ)≡κ_(rr)−κ₀₀ is given by $\begin{matrix}{\kappa_{\Delta} = {{- 6}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}^{2}}\left\{ \begin{matrix}{{\left( {4 - {3\frac{h_{s}}{h_{s\quad 0}}}} \right)\left( {ɛ_{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}\quad{\mathbb{d}\eta}}}}} \right)} +} \\{{\left( {\frac{h_{s}}{h_{s\quad 0}} - 1} \right)ɛ_{m}} - {\frac{2}{r^{2}}{\int_{0}^{r}{{\eta\left\lbrack {\frac{h_{s}(\eta)}{h_{s\quad 0}} - 1} \right\rbrack}{ɛ_{m}(\eta)}\quad{\mathbb{d}\eta}}}} -} \\{\frac{1}{r^{2}}{\int_{0}^{r}{{\eta^{2}\left\lbrack {{ɛ_{m}(\eta)} + {\frac{3\left( {1 + v_{s}} \right)}{\eta^{2}}{\int_{0}^{\eta}{{{\rho ɛ}_{m}(\rho)}{\mathbb{d}\rho}}}} + {\frac{3\left( {1 - v_{s}} \right)}{2}\overset{\_}{ɛ_{m}}}} \right\rbrack}\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}\quad{\mathbb{d}\eta}}}}\end{matrix}\quad \right\}}} & (4.2)\end{matrix}$

The thin film stresses are obtained from the constitutive relations$\sigma_{rr}^{(f)} = {{\frac{E_{f}}{1 - v_{f}^{2}}\left\lbrack {u_{f}^{\prime} + {v_{f}\frac{u_{f}}{r}} - {\left( {1 + v_{f}} \right)ɛ_{m}}} \right\rbrack}\quad{and}}$${\sigma_{\theta\theta}^{(f)} = {\frac{E_{f}}{1 - v_{f}^{2}}\left\lbrack {{v_{f}u_{f}^{\prime}} + \frac{u_{f}}{r} - {\left( {1 + v_{f}} \right)ɛ_{m}}} \right\rbrack}},$where u_(f) is given in (2.9). The sum of thin film stresses, up to theO(β²) accuracy (as compared to unity), is related to the misfit strainby $\begin{matrix}{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{f}}{1 - v_{f}}{\left( {{- 2}ɛ_{m}} \right).}}} & (4.3)\end{matrix}$

The difference of thin film stresses σ_(rr) ^((f))−σ_(θθ) ^((f)) is onthe order of${O\left( {\frac{E_{f}^{2}}{E_{s}}ɛ_{m}\frac{h_{f}}{h_{s\quad 0}}} \right)},$which is very small as compared to σ_(rr) ^((f))+σ_(θθ) ^((f)).Therefore only its leading term is presented $\begin{matrix}{{\sigma_{r\quad r}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {4E_{f}\frac{E_{f}h_{f}}{1 - v_{f}^{2}}{{\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}}\left\lbrack {ɛ_{m} - {\frac{2}{r^{2}}{\int_{0}^{r}{{{\eta ɛ}_{m}(\eta)}{\mathbb{d}\eta}}}}} \right\rbrack}.}}} & (4.4)\end{matrix}$5.3.1. Special Case: Uniform Misfit Strain

For uniform misfit strain distribution ε_(m)=constant (and non-uniformsubstrate thickness), the interface shear stress in (3.3) vanishes. Thecurvatures in (4.1) and (4.2) become${\kappa_{\Sigma} = {{- 12}\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}}{E_{s}h_{s\quad 0}^{2}}\left\{ {1 - {\frac{5 - v_{s}}{2}\left( {\frac{h_{s}}{h_{s\quad 0}} - 1} \right)} + {\left( {1 - {2v_{s}}} \right)\frac{h_{s} - {h_{s}(0)}}{h_{s\quad 0}}}} \right\} ɛ_{m}}},\quad{\kappa_{\Delta} = {18\frac{E_{f}h_{f}}{1 - v_{f}}\frac{1 - v_{s}^{2}}{E_{s}h_{s\quad 0}^{2}}\left\{ {\frac{h_{s}}{h_{s\quad 0}} - {\frac{2}{r^{2}}{\int_{0}^{r}{\eta\frac{h_{s}(\eta)}{h_{s\quad 0}}{\mathbb{d}\eta}}}}} \right\}{ɛ_{m}.}}}$It is clear that the curvatures are only equi-biaxial (i.e., κ_(Δ)=0)for the uniform substrate thickness.

The thin film stresses are equal-biaxial for uniform misfit strain.Elimination of uniform misfit strain from the above κ_(Σ) and (4.3)gives the extended Stoney formula for non-uniform substrate thickness,$\begin{matrix}{\sigma_{rr}^{(f)} = \sigma_{\theta\theta}^{(f)}} \\{{= {\frac{E_{s}h_{s\quad 0}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\left\{ {1 + {\frac{5 - v_{s}}{2}\left( {\frac{h_{s}}{h_{s\quad 0}} - 1} \right)} - {\left( {1 - {2v_{s}}} \right)\frac{h_{s} - {h_{s}(0)}}{h_{s\quad 0}}}} \right\}\frac{\kappa_{\Sigma}}{2}}},}\end{matrix}$where we have used |h_(s)−h_(s0)|<<h_(s0). The above relation can alsobe rewritten as $\begin{matrix}{\sigma_{rr}^{(f)} = \sigma_{\theta\theta}^{(f)}} \\{= {\frac{E_{s}h_{s}^{2}}{6\left( {1 - v_{s}} \right)h_{f}}\left\{ {1 + {\frac{1 - v_{s}}{2}\left( {\frac{h_{s}}{h_{s\quad 0}} - 1} \right)} - {\left( {1 - {2v_{s}}} \right)\frac{h_{s} - {h_{s}(0)}}{h_{s\quad 0}}}} \right\}\frac{\kappa_{\Sigma}}{2}}}\end{matrix}$via the Taylor expansion$h_{s\quad 0}^{2} = {{h_{s}^{2}\left\lbrack {1 - {2\left( {\frac{h_{s}}{h_{s\quad 0}} - 1} \right)} + {O\left( \beta^{2} \right)}} \right\rbrack}.}$Only for uniform substrate thickness the above formula degenerates tothe Stoney formula in (1.1). In the following, we extend such a relationfor arbitrary non-uniform misfit strain distribution and non-uniformsubstrate thickness.5.4 Extension of Stoney Formula for Non-Uniform Misfit StrainDistribution and Non-Uniform Substrate Thickness

In this section we extend the Stoney formula for arbitrary non-uniformmisfit strain distribution and non-uniform substrate thickness byestablishing the direct relation between the thin-film stresses andsubstrate curvatures. We invert the misfit strain from (4.1) as$\begin{matrix}{{ɛ_{m} = {{- \frac{1 - v_{f}}{6E_{f}h_{f}}}\frac{E_{s}}{1 - v_{s}^{2}}\begin{Bmatrix}{{h_{s}^{2}\kappa_{\Sigma}} - {\frac{1 - v_{s}}{2}\overset{\_}{h_{s}^{2}\kappa_{\Sigma}}} +} \\{{\frac{1}{2}{\int_{r}^{R}{\begin{bmatrix}{{\left( {1 - {3\quad v_{s}}} \right){\kappa_{\Sigma}(\eta)}} -} \\{3\left( {1 - v_{2}} \right){\kappa_{\Delta}(\eta)}}\end{bmatrix}{h_{s}^{2}(\eta)}\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{\mathbb{d}\eta}}}} -} \\{\frac{1 - v_{s}}{R^{2}}{\int_{0}^{R}{{\eta^{2}\left\lbrack {{\kappa_{\Sigma}(\eta)} - {\kappa_{\Delta}(\eta)}} \right\rbrack}{h_{s}^{2}(\eta)}\frac{h_{s}^{\prime}(\eta)}{h_{s\quad 0}}{\mathbb{d}\eta}}}}\end{Bmatrix}}},} & (5.1)\end{matrix}$where$\overset{\_}{h_{s}^{2}\kappa_{\Sigma}} = {\frac{2}{R^{2}}{\int_{0}^{R}{\eta\quad h_{s}^{2}\kappa_{\Sigma}{\mathbb{d}\eta}}}}$is the average of h_(s) ²κ_(Σ), and we have used (4.2) in establishing(5.1).

The thin film stresses are obtained by substituting (5.1) into (4.3) and(4.4) as $\begin{matrix}{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}}{3\left( {1 - v_{s}^{2}} \right)h_{f}}\begin{Bmatrix}{{h_{s}^{2}\kappa_{\Sigma}} - {\frac{1 - v_{s}}{2}\overset{\_}{h_{s}^{2}\kappa_{\Sigma}}} +} \\{{\frac{1}{2}{\int_{r}^{R}{\begin{bmatrix}{{\left( {1 - {3v_{s}}} \right){\kappa_{\Sigma}(\eta)}} -} \\{3\left( {1 - v_{s}} \right){\kappa_{\Delta}(\eta)}}\end{bmatrix}{h_{s}^{2}(\eta)}\frac{h_{s}^{\prime}(\eta)}{h_{{s\quad 0}\quad}}{\mathbb{d}\eta}}}} -} \\{\frac{1 - v_{s}}{R^{2}}{\int_{0}^{R}{{\eta^{2}\left\lbrack {{\kappa_{\Sigma}(\eta)} - {\kappa_{\Delta}(\eta)}} \right\rbrack}{h_{s}^{2}(\eta)}\frac{h_{s}^{\prime}}{h_{s\quad 0}}{\mathbb{d}\eta}}}}\end{Bmatrix}}},} & (5.2) \\{\quad{{\sigma_{rr}^{(f)} - \sigma_{\theta\theta}^{(f)}} = {{- \frac{2E_{f}h_{s\quad 0}}{3\left( {1 + v_{f}} \right)}}{\kappa_{\Delta}.}}}} & (5.3)\end{matrix}$

Equations (5.2) and (5.3) provide direct relations between film stressesand system curvatures. The system curvatures in (5.2) always appeartogether with the square of substrate thickness, i.e., h_(s) ²κ_(Σ) andh_(s) ²κ_(Δ). Stresses at a point in the thin film depend not only oncurvatures at the same point (local dependence), but also on curvaturesin the entire substrate (non-local dependence) via the term h_(s) ²κ_(Σ)and the integrals in (5.2). For uniform substrate thickness, Eqs. (5.2)and (5.3) degenerate to the equations previously described for theuniform substrate thickness.

The interface shear stress r can also be directly related to systemcurvatures via (3.3) and (5.1) $\begin{matrix}{\tau = {\frac{E_{s}}{6\left( {1 - v_{s}^{2}} \right)}{\left\{ {{\frac{\mathbb{d}}{\mathbb{d}r}\left( {h_{s}^{2}\kappa_{\Sigma}} \right)} - {{\frac{1}{2}\begin{bmatrix}{{\left( {1 - {3v_{s}}} \right)h_{s}^{2}\kappa_{\Sigma}} -} \\{3\left( {1 - v_{s}} \right)h_{s}^{2}\kappa_{\Delta}}\end{bmatrix}}\frac{h_{s}^{\prime}}{h_{s\quad 0}}}} \right\}.}}} & (5.4)\end{matrix}$

Equation (5.4) provides a way to determine the interface shear stressesfrom the gradients of system curvatures once the full-field curvatureinformation is available. Since the interfacial shear stress isresponsible for promoting system failures through delamination of thethin film from the substrate, Eq. (5.4) has particular significance. Itshows that such stress is related to the gradient of κ_(rr)+κ_(θθ) aswell as to the magnitude of κ_(rr)+κ_(θθ) and κ_(rr)−κ_(θθ) fornon-uniform substrate thickness.

In summary, (5.2)-(5.4) provide a simple way to determine the thin filmstresses and interface shear stress from the non-uniform misfit strainin the thin film and non-uniform substrate thickness.

5.5. Thin Film/Substrate Systems With Non-Uniform Film Thickness andNon-Uniform Substrate Thickness

For a thin film/substrate system with uniform substrate thickness butnon-uniform film thickness, the thin-film stress is inverselyproportional to the film thickness, i.e., $\begin{matrix}{{{\sigma_{rr}^{(f)} + \sigma_{\theta\quad\theta}^{(f)}} = {\frac{E_{s}h_{s}^{2}}{6{h_{f}\left( {1 - v_{s}} \right)}}\begin{bmatrix}{\kappa_{rr} + \kappa_{\theta\theta} + {\frac{1 - v_{s}}{1 + v_{s}}\left( {\kappa_{rr} + \kappa_{\theta\theta} - \overset{\_}{\kappa_{rr} + \kappa_{\theta\theta}}} \right)} +} \\{\frac{1 - v_{s}}{1 + v_{s}}{\int\limits_{A}{\ldots\quad\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right){\mathbb{d}A}}}}\end{bmatrix}}},} & (5.5)\end{matrix}$where the film thickness h_(f) must be taken as its local, non-uniformvalue; κ_(rr)+κ_(θθ) is the average of κ_(rr)+κ_(θθ) over the entirefilm A; the coefficient is independent of the (non-uniform) filmthickness and depends only on positions in the film; and the integrationis over the entire film.

For a thin film/substrate system with non-uniform substrate thicknessbut uniform film thickness, the thin-film stress can be written as:$\begin{matrix}{{{\sigma_{rr}^{(f)} + \sigma_{\theta\theta}^{(f)}} = {\frac{E_{s}}{6{h_{f}\left( {1 - v_{s}} \right)}}\begin{Bmatrix}{{h_{s}^{2}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)} +} \\{{\frac{1 - v_{\quad s}}{\quad{1 + v_{\quad s}}}\left\lbrack {{h_{\quad s}^{\quad 2}\left( {\kappa_{\quad{rr}} + \kappa_{\quad{\theta\theta}}} \right)} - \overset{\_}{h_{s}^{2}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)}} \right\rbrack} +} \\{{\frac{1 - v_{s}}{1 + v_{s}}{\int\limits_{A}{\ldots\quad{h_{s}^{2}\left( {\kappa_{rr} + \kappa_{\theta\theta}} \right)}{\mathbb{d}A}}}} +} \\{\int\limits_{A}{\left( {{\ldots\quad\frac{\partial\quad h_{\quad s}}{\partial r}} + {\ldots\quad\frac{\partial\quad h_{\quad s}}{\partial\theta}}} \right){\mathbb{d}A}}}\end{Bmatrix}}},} & (5.6)\end{matrix}$where the square of substrate thickness h_(s) ² appears together withthe curvatures, and the last term represents the contribution fromnon-uniform substrate thickness.

For a thin film/substrate system with both non-uniform substratethickness and non-uniform film thickness, the above formula (5.6) stillholds except that the film thickness h_(f) should be taken as its localvalue as in formula (5.5).

In implementations, the above described techniques and their variationsmay be implemented as computer software instructions. Such softwareinstructions may be stored in an article with one or moremachine-readable storage media or stored in one or more machine-readablestorage devices connected to one or more computers as part of theprocessor shown in FIG. 1A. In operation, the instructions are executedby, e.g., one or more computer processors, to cause the machine (e.g.,the processor in FIG. 1A) to perform the described functions andoperations.

In general, embodiments of the invention and all of the functionaloperations described in this specification can be implemented in digitalelectronic circuitry, or in computer software, firmware, or hardware,including the structures disclosed in this specification and theirstructural equivalents, or in combinations of one or more of them.Embodiments of the invention can be implemented as one or more computerprogram products, i.e., one or more modules of computer programinstructions encoded on a computer readable medium for execution by, orto control the operation of, data processing apparatus. The computerreadable medium can be a machine-readable storage device, amachine-readable storage substrate, a memory device, a composition ofmatter effecting a machine-readable propagated signal, or a combinationof one or more them. The term “data processing apparatus” encompassesall apparatus, devices, and machines for processing data, including byway of example a programmable processor, a computer, or multipleprocessors or computers. The apparatus can include, in addition tohardware, code that creates an execution environment for the computerprogram in question, e.g., code that constitutes processor firmware, aprotocol stack, a database management system, an operating system, or acombination of one or more of them. A propagated signal is anartificially generated signal, e.g., a machine-generated electrical,optical, or electromagnetic signal, that is generated to encodeinformation for transmission to suitable receiver apparatus.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a stand alone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network. The processes and logic flowsdescribed in this specification can be performed by one or moreprogrammable processors executing one or more computer programs toperform functions by operating on input data and generating output.

Referring back to FIG. 1A, the above sections provide detaileddescriptions on the processing algorithms for the processor thatperforms the stress analysis based on the full-field curvature map ofthe layered plate structure under measurement. The full-field curvaturemap may be obtained by various techniques. Non-invasive, full-fieldoptical methods may be preferred since such methods are efficient andfast and may allow for real time and in situ monitoring of substratesunder fabrication.

The following sections describe examples of such optical probing methodsfor measuring the full-field curvature map of the layered structureunder test based on optical interferometry. Such optical methods can beused to achieve non-invasive, full-field measurements of patterned andunpatterned surfaces in various devices and structures. Describedimplementations include measurements of patterned and un-patternedsurface profiles of various surfaces by using optical shearinginterferomety. Optical interferometry techniques for illumination of anun-patterned backside surface of a wafer or substrate with a patternedfront surface are also described. When properly configured, a surfacemonitoring system based on one of the disclosed optical techniques mayprovide full-field measurements of a surface in real time. In addition,such a surface monitoring system may provide in-situ monitoring of awafer under processing.

As an example, a method for monitoring a substrate fabrication processbased on the techniques described in this application may include thefollowing. When a layered structure is being processed, an optical probeis directed to the layered structure to optically obtain a full-fieldcurvature map of the layered structure. The full-field curvature map isprocessed to obtain curvature information at all locations of thelayered structure. Next, the total stresses at each location of thelayered structure can be determined based on the present techniques.This stress analysis can be performed by a computer and may be conductedin real time during the fabrication process. When the stress at one ormore locations exceeds an acceptable threshold value, the layeredstructure may be defective and should be flagged or removed fromsubsequent fabrication steps. The interface shear stress, for example,can be monitored to determine whether a film formed on a substrate maybe delaminated due to the stresses.

The optical probing for obtaining the full-field measurement of thelayered structure may be implemented in various implementations. In oneimplementation, for example, an optical probe beam with a substantiallyuniform wavefront may be used to illuminate a surface under measurementto produce a reflected probe beam with a reflected wavefront thatcarries distortions caused by an illuminated area on the surface. Thereflected probe beam is directed through an optical shearinginterferometer device to obtain an optical interference pattern betweenthe reflected wavefront and another replica of the reflected wavefrontthat is spatially shifted by a shearing distance. Next, a phase shiftbetween the reflected wavefront and the replica of the reflectedwavefront is adjusted to obtain a plurality of phase-shiftedinterference patterns of different phase shifts from the opticalshearing interferometer. The interference patterns are then processed toobtain information on surface slopes across the illuminated area in thesurface under measurement.

In other implementations, support members may be to contact a backsidesurface of a wafer to hold the wafer (or of a reticle to hold thereticle, etc.). The wafer is fabricated with patterns on a front surfaceopposite to the backside surface. The backside surface is illuminatedwith a probe beam to produce a reflected probe beam with a reflectedwavefront that carries distortions caused by an illuminated area on thebackside surface. An optical interference pattern is then produced withthe reflected probe beam to include discontinuities due to presence ofsupport members on the backside surface. An interpolation algorithm isapplied in processing the optical interference pattern to interpolateinterference fringes caused by the backside surface across regions withthe discontinuities to obtain interference pattern features within theilluminated area that are caused solely by the backside surface. Next,the interpolated interference pattern from the backside surface areprocessed to obtain surface slopes of corresponding positions on thefront surface of the wafer.

Alternatively, the above interpolation in data processing may besubstituted by additional measurements and processing to obtain data inthe areas on the backside surface occupied by the support members. Forexample, without applying the interpolation, the interference patternfrom the backside surface is processed to obtain surface slopes ofcorresponding positions on the front surface of the wafer. Next, theangular orientation of the wafer on the support members is changed atleast once to obtain at least one another reflected optical probe beamfrom the same incident optical probe beam and thus another opticalinterference pattern. The other interference pattern from the backsidesurface is then processed to obtain surface slopes of correspondingpositions on the front surface of the wafer. The surface slopes obtainedfrom different interference patterns at different angular orientationsof the wafer are then compared. The missing data found at a location inone interference pattern is filled by data at the same location inanother interference pattern obtained at a different angularorientation.

This application also describes techniques for using interferencepatterns obtained at different shearing distances in a shearinginterferometer to improve the measurements. In one implementation, forexample, an optical probe beam with a substantially uniform wavefront isused to illuminate a surface under measurement to produce a new opticalbeam with a distorted wavefront caused by the surface. The new opticalbeam is directed through an optical shearing interferometer to obtain anoptical interference pattern between the distorted wavefront and anotherreplica of the distorted wavefront that is spatially shifted by ashearing distance. The shearing distance is then adjusted to obtainoptical interference patterns at different shearing distances. Theseinterference patterns at different shearing distances are processed toextract information on the surface under measurement.

In the above example, two interference patterns with two differentshearing distances may be subtracted to produce a differentiatedinterference pattern that corresponds to a new shearing distance equalto a difference between the two different shearing distances. Thistechnique can be used to obtain data at a small shearing distance thatmay be difficult to achieve with the given shearing interferometer.

This application further describes a number of shearing interferometersdifferent from a coherent gradient sensing (CGS) system for measuringsurfaces. These non-CGS shearing interferometers may have certainadvantages over CGS in specific applications.

Optical shearing interferometers produce and interfere two spatiallyshifted replicas of the same, usually distorted wavefront of an opticalbeam along a direction transverse to the direction of propagation of thethe wavefront. For example, transverse and radial shearinginterferometers may be used. The interference between the spatiallyshifted replicated wavefronts generates an interference patternrepresenting the spatial distribution of slopes in the wavefront. Ineffect such interferometers perform an optical differentiation of thewavefront. In some of the examples for optically measuring surfacesdescribed in this application, at least one optical shearinginterferometer may be used to optically measure a surface byilluminating the surface with a collimated probe beam. The shearinginterferometer may be configured to produce a shearing interferencepattern from either of the optical transmission of the probe beamthrough the surface or from the optical reflection of the probe beam bythe surface. The shearing interference pattern is then processed toobtain surface, slopes, curvatures and other surface topographicalinformation. For example, surface topography on the global profile ofthe surface and nanotopography on the local profile of the surface maybe obtained from the shearing interferometer. Examples of measurablesurfaces include but are not limited to surfaces in various panels andplates, various substrates and wafers, integrated electronic circuits,integrated optical devices, opto-electronic circuits, andmicro-electro-mechanical systems (MEMs), flat panel display systems(e.g., LCD and plasma displays), and photolithography masks, pelliclesand reticles.

The use of optical shearing interferometry presents certain advantagesin optically measuring surfaces. Optical hearing interferometry may bean effective measuring tool for surfaces patterned with variousmicrostructures such as patterned wafers and patterned mask substrates.In addition, an optical shearing interferometer may be used for in-situmonitoring of the surface properties such as curvatures and relatedstresses during fabrication of devices at the wafer level and themeasurements may be used to dynamically control the fabricationconditions or parameters in real time. As an example, measurement andoperation of an optical shearing interferometer generally is notsignificantly affected by rigid body translations and rotations due tothe self-referencing nature of the optical shearing interferometry.Hence, a wafer or device under measurement may be measured by directinga probe beam substantially normal to the surface or at low incidentangles without affecting the measurements. By shifting or shearing thewavefront, the optical shearing interferometer measures the deformationof one point of the wavefront to another separated by the shearingdistance, i.e., the distance between the two interfering replicas of thesame wavefront. In this sense, the optical shearing interferometer isself referencing and thus increases its insensitivity or immunity tovibrations of the wafer or device under measurement. This resistance tovibrations may be particularly advantageous when the measurement isperformed in a production environment or in situ during a particularprocess (e.g. deposition within a chamber), where vibration isolation isa substantial challenge.

As a comparison, many non-shearing interferometers generate wavefrontinterference of topology or topography (surface elevation) based onoptical interference between a distorted wavefront reflected from asample surface and an undistorted, reference wavefront reflected from aknown reference surface. The use of such non-shearing opticalinterferometers for measuring patterned surfaces may be ineffectivebecause, in many cases, the relatively non-uniform or diffuse wavefrontreflected off the patterned surface may not interfere with fidelity withthe wavefront reflected off the reference surface due to, e.g., the lossof light diffracted or scattered by the pattern into large angles. Also,a patterned surface may have substantially different reflectiveproperties, e.g., certain areas of the patterned surface are highlyabsorbing at the probe wavelength than other areas of the patternedsurface or the reference surface. In these and other circumstances, itmay be difficult to unwrap and interpret the interferometric imagesproduced by such non-shearing interferometers in the presence ofextensive patterning.

Another feature of the shearing interferometry is that the wavefront isoptically differentiated once and the optical differentiation isrecorded in the shearing interference pattern. Hence, only a singlederivative operation on the data from the shearing interference patternis sufficient to calculate curvatures from slopes of the wavefront. Thisreduces the amount of computation in processing the interference dataand thus reduces the time for data processing. Also, because theshearing interferometry method provides full-field interferometric datait can utilize many more data points compared to other methods such asthe method of using a conventional capacitive probe to measure a fewpoints (e.g., 3 points) of surface topology. This higher data densityprovides more accurate measurements and better resistance to noise thanother methods which feature much less density of measured data. Inaddition, although various laser beam scanning tools may be used tomeasure wafer bow or surface curvature, these methods typically measureradial curvature only. Shearing interferometry may be easily implementedto measure surface slopes in two orthogonal directions (X and Y) withinthe surface and thus allow for elucidation of the full curvature tensorand the associated stress states of the wafer or substrate.

In applying shearing interferometry for measuring patterned surfaces onwafers and other structures (e.g. patterned mask elements), thepatterned wafers, e.g., semiconductor and optoelectronic wafers withdiameters of 200 mm, 300 mm or other wafer sizes may be placed in ashearing interferometer in a configuration that allows a collimatedprobe beam to be reflected off the wafer surface. The shearinginterferometer uses the reflected probe beam from the wafer surface toproduce the two interfering wavefronts, which are substantially similarin shape after being sheared by a small shearing distance. Hence, theinterference between the two wavefronts produces coherent interference.Although each wavefront reflected off a patterned surface may beinherently noisy and diffuse, sufficient coherence exists between thewavefronts to produce meaningful fringe patterns and can be interpretedto extract surface information.

FIG. 1 illustrates one implementation of a system 100 for measuring aspecimen surface 130 based on optical shearing interferometry. A lightsource 110 is provided to produce a collimated probe beam 112 with asubstantially uniform wavefront. The light source 110 may produceradiation in a wide range spectral ranges including visible andinvisible wavelengths (e.g., IR and UV radiation). The light from thesource 110 may be coherent. Some interferometers, such as a CGS device,may also operate with incoherent light. This probe beam 112 is directedto illuminate the surface 130 and to produce a reflected probe beam 132.An optical element 120 such as a beam splitter may be used to direct theprobe beam 110 to the surface 130 and to transmit the reflected probebeam 132. A shearing device 101, i.e., an optical shearinginterferometer, is placed in the optical path of the reflected probebeam 132 to generate a shearing interference pattern from the reflectedprobe beam 132. Oblique incidence of the collimated beam 112 onto thereflective surface 130 may also be used and the beamsplitter element 120is bypassed. In general, any shearing interferometer may be used toimplement the shearing device 101. In actual applications, differentshearing configurations may have unique features or attributes and henceare different from one another in this context. Examples of the shearingdevice 101 include a coherent gradient sensing (CGS) system usingoptical gratings to cause the shearing of the wavefront, a radial shearinterferometers, wedge plate in a Bi-Lateral Shearing Interferometer(U.S. Pat. No. 5,710,631), and others, some of which are described inlater sections of this application.

The system 100 also includes a light collecting unit 102 in the outputoptical path of the shearing device 101 to direct the optical output ofthe shearing device 101, the shearing interference pattern, to animaging sensor 180, such as a camera (e.g., a CCD or other pixel sensingarray). The light collecting unit 102 may include a filtering lens 160and a spatial filter plane 170 when the shearing device 101 uses opticalgratings to produce the shearing. The imaging sensor 180 converts theshearing interference pattern into an electronic form and a signalprocessing circuit which may include a computer is used to process theshearing interference pattern to extract desired surface information.

The measurements of patterned wafers by optical reflection with ashearing interferometer may be improved by using phase shifting in themeasurements. Phase shifting may be implemented to progressively adjustthe phase separation between the two shifted interfering wavefrontswhich cycles or manipulates fringe position on the specimen's surfaceunder measurement. In one implementation, a shearing interferometer maybe configured to obtain multiple phased images of a patterned wafer'ssurface, for example at 0, 90, 180, 270 and 360 degrees in phase. Thephase shifting method allows for the wavefront slope to be measured bycalculating the “relative phase” modulation at each pixel on a detectorarray that receives the interference pattern. The phase shifting methodalso allows for consistent interpretation of wavefront and specimenslope on a surface that exhibits changing reflectivity, like those foundon patterned wafers. On a patterned wafer surface, each pixel locationon the specimen may reflect light with a varying degree of intensityrelative to other pixel locations. This may complicate theinterpretation of any single shearing interferogram. The phase shiftingmethod in shearing interferometry can simultaneously increase theaccuracy of the slope resolution and allow for accurate interpretationof interferograms on patterned surfaces with a spatially varying opticalreflectivity. This is possible in part because the relative phase ofeach pixel or location within the shearing interfering pattern ismeasured, rather than merely measuring the variation in the fringeintensity.

FIG. 2 illustrates an example of a measurement of the relative phase inthe phase-shifted interference pattern. The interference pattern imageon the left was collected off the patterned surface of a 300 mm siliconwafer. The interference pattern represents one out a series of, forexample, 5 phase-shifted interference patterns. The detail in the upperright portion of the image illustrates that locally, the fringeintensity may vary dramatically from point to point as a result of thepatterning on the wafer surface. For comparison, a fringe pattern on abare or continuous surface would have smooth and continuous variationsin fringe intensity. The inserted graph in the bottom right of FIG. 2shows schematically the variation in the fringe intensity as a functionof the phase-shift value or angle for two points on the patterned wafersurface. The phase axis has angular increments of 90 degrees, whereasthe intensity axis is meant to represent the full dynamic range of theCCD or other imaging array. Point 1 marked by the arrow on the leftcorresponds to a pixel region on the wafer where the reflectivity isrelatively high and is indicated schematically by the large amplitude ofthe curve. Point 2 marked by the arrow on the right corresponds to apixel region on the wafer where the reflectivity is relatively low andis indicated schematically by the smaller amplitude curve. When phaseshifting is implemented, the relevant quantity of interest is therelative phase angle, or the horizontal offset of one curve (e.g. Point1 curve) relative to the other (e.g. Point 2 curve), and not theamplitude of the curve. The intensity at any given point from a seriesof phase shifted interferograms should be sufficiently large to enableadequate characterization of the relative phase offset.

In implementation of the phase shifting, the collected multiplephase-shifted interferograms of the patterned wafer surface aresubsequently processed by a phase extraction algorithm and an unwrappingalgorithm to accurately interpret the surface slopes embedded in thephase-shifted interferograms. Suitable phase extraction algorithms mayinclude Bucket nA, nB, or nC, where ‘n’ is the number of frames in thephase shifted data set. Phase extraction algorithms other than the aboveBucket A type, Bucket B type, and Bucket C type algorithms may also beused. Suitable unwrapping algorithms may include Minimum Discontinuity(MDF) and Preconditioned Conjugate Gradient (PCG) algorithms. Inaddition, Branch Cut Minimization and Tiled Modulation Guided algorithmsmay also be used to process the phase-shifted interferograms and may beeffective in unwrapping lightly patterned surfaces.

Once the phase-shifted interferograms have been unwrapped, theinterpretation of raw slope data and the derivation of curvature may beenhanced by statistically fitting a surface polynomial to the raw slopedata. Statistical surface fits, including Zernicke polynomials andLegendre polynomials, may be applied to raw slope data derived fromPatterned Wafers for the purpose of deriving topography (ornanotopography) and curvature data.

One property of the shearing interferometry due to its self-referencingnature is that the resulting shearing interference pattern essentiallymeasures the deviations from flatness of the surface under measurementby using the surface itself as a reference surface. Such relative dataon surface height or flatness may be useful in various applicationswhere the height or flatness of a surface is monitored or controlled.For example, in a chemical mechanical polishing (CMP) process or othersurface polishing processes, the relative height across the surface maybe monitored to determine the effectiveness of the polishing process. Ashearing interferometer may be used to monitor the surface flatness andthe measurements may be used to dynamically control the polishingcondition of the polishing process in real time.

In some implementations, the shearing distance between the transverselyshifted wavefronts that interfere with each other may be adjusted duringthe measurement process to improve the resolution and accuracy of thedata. By capturing interferometric images of the surface at multipleincrements of shearing distances, it is possible to resolve featuressmaller than the effective pixel size of the camera or imaging sensingarray being used to sample the interferometric data. In addition, asdescribed later in this application, the use of multiple shearingdistances enables the highly accurate calculation of the estimatedsurface topography or nanotopography from the relative data by ageometric calculation rather than a standard numerical integrationalgorithm to compute the actual surface profile.

Referring back to FIG. 1, the system 100 may be used to measure surfacesof a variety of wafers, substrates, flat panels or lithographic maskelements. The system 100 can simultaneously measure each and every pointin the illuminated area on the specimen surface to obtain information onthe flatness, topography, slope, curvature and stress. The shearinginterferometry may be especially advantageous in measuring patternedsurfaces such as microfabricated surfaces commonly found insemiconductor or optoelectronics wafers and substrates. The shearingdevice 101 may produce coherent or semi-coherent interference on apatterned surface.

As an example, FIG. 3 shows an exemplary implementation of a coherentgradient sensing (“CGS”) system 300 based on the system design inFIG. 1. Certain aspects of the system 300 are described in U.S. Pat. No.6,031,611 to Rosakis et al., which is incorporated herein by reference.The CGS system 300 uses a collimated coherent optical beam 112 from alight source 110 as an optical probe to obtain surface slope andcurvature information indicative of a specularly reflective surface 130formed of essentially any material. An optical element 120 such as abeam splitter can be used to direct the beam 112 to the surface 130.When the reflective surface 130 is curved, the wavefront of thereflected probe beam 132 is distorted and thereby the reflected probebeam 132 acquires an optical path difference or phase change associatedwith the surface topographic of the surface 130 under measurement. Thissystem produces a “snapshot” of each point within the illuminated areaon the surface 130 and hence the surface topographic information at anypoint along any direction within the illuminated area can be obtained.This can eliminate the need for measuring one point at a time in asequential manner by scanning a probe beam one point at a time acrossthe surface 130.

Two optical diffraction elements such as optical diffraction gratings140 and 150 spaced from each other by Δ are placed in the path of thereflected probe beam 132 to manipulate the distorted wavefront forcurvature measurement. Two diffraction components produced by the secondgrating 150 from diffraction of two different diffraction componentsproduced by the first grating 140 are combined, by using an opticalelement 160 such as a lens, to interfere with each other. When a lens isused as the optical element, the two diffracted beams produced by thesecond grating 150 and combined by the lens have the same diffractionangle out of the second grating 150 and thus are parallel to each other.The diffraction by the two gratings 140 and 150 effectuates a relativespatial displacement, i.e., a lateral spatial shift, between the twoselected diffraction components. This shift is a function of the spacingΔ between the two gratings 140 and 150 when other grating parameters arefixed. More specifically, the shearing distance is (Δ×tan θ), where θ isthe diffraction angle of two interfering diffraction beams. Hence, thegratings 140 and 150 produce two spatially shifted wavefronts from thesame wavefront of the reflected probe beam 132. A spatial filter 170 isplaced relative to the optical element 160 to transmit the interferencepattern of the selected diffraction components and to block otherdiffraction orders from the second grating 150. In general, any desireddiffraction order or combination of orders may be selected for themeasurements.

The transmitted interference pattern is then captured by an imagingsensor 180 which may include an array of sensing pixels, such as a CCDarray, to produce an electrical signal representing the interferencepattern. A signal processor 190, processes the electrical signal toextract a spatial gradient of the wavefront distortion caused by thetopography of the reflective surface 130. This spatial gradient, inturn, can be further processed to obtain the curvature information andhence a curvature map of the illuminated area on the surface 130 can beobtained. A single spatial differentiation is performed on theinterference pattern to measure the surface gradient. This technique canprovide accurate measurements of surface curvatures and the accuracy ishigh when the curvature variation of the surface is gradual, i.e., whenthe out-of-plane displacement is less than the thickness of the film,the line or the substrate. This technique is insensitive to rigid bodymotions in contrast to some other interferometric techniques. Details ofthis data processing operation are described in the above-referencedU.S. Pat. No. 6,031,611 to Rosakis et al. Upon completing the processingfor the surface slopes and curvatures, the processor 190 furtheroperates to compute the stresses from the surface curvatures.

The two gratings 140 and 150 in general may be any gratings, withdifferent grating periods and oriented with respect to each other at anyangle. Preferably, the two gratings may be oriented with respect to eachother in the same direction and may have the same grating periods tosimplify the data processing. In this case, the grating direction isessentially set by the direction of the relative spatial displacement(“shearing”) between the two selected diffraction components due to thedouble diffractions by the gratings 140 and 150.

Certain applications may require spatial shearing in two differentdirections to obtain a full-field two-dimensional slope and curvaturemeasurement. This may be done by using the CGS system 300 to perform afirst measurement when the sample surface 130 is at a first orientationand subsequently to perform a second measurement when the sample surface130 is rotated to a second orientation (e.g., perpendicular to the firstorientation).

Alternatively, a two-arm CGS system, shown in FIG. 4 may be implementedto have two separate sets of double gratings in two different directionsto simultaneously produce the interference pattern in two differentspatial shearing directions. Hence, time-varying effects in thetopography, slope and curvature distribution in both spatial shearingdirections can be obtained.

In addition, each of the two gratings 140 and 150 in FIG. 3 may bereplaced by a grating plate with two orthogonal cross gratings toeffectuate the two dimensional shearing of the system in FIG. 4. Thespatial filter 170 may be replaced by a substitute filter with anadditional optical aperture shifted along the direction of x1 toselectively transmit an interference pattern for shearing along theorthogonal direction.

In the above exemplary CGS systems, the phase shifting may be achievedby changing the relative position between the two gratings 140 and 150.In one implementation, the relative position of the two gratings 140 and150 in the transverse plane defined by directions x1 and x2 may beadjusted while maintaining the spacing between the two gratings 140 and150 along the x3 direction fixed at a desired constant. FIG. 5Aillustrates a CGS system where a positioning mechanism, such as precisetranslation stage or a positioning transducer, is used to implement thisadjustment of the relative position between the gratings for phaseshifting. At least one lateral position controller may be engaged to oneof the two gratings to cause the lateral change in position. Two lateralposition controllers may be respectively engaged to the two gratings 140and 150 to cause the phase shift. In this implementation, the twogratings may be maintained to be parallel to each other with the fixedspacing during the lateral movement. Multiple shearing interferencepatterns with different lateral relative positions between the gratings140 and 150 can be obtained for further processing with phase extractionand unwrapping algorithms.

FIG. 5B shows another way for implementing the phase shifting mechanismin CGS. In this configuration, the relative lateral position between thetwo gratings 140 and 150 is fixed and the two gratings 140 and 150 aremaintained to be substantially parallel. A position control mechanism isimplemented to slightly change the spacing, Δ, between the two gratings140 and 150 along the x3 direction by a small amount (δ). The magnitudeof δ is much less than the desired spacing Δ so the spacing Δ and themeasurement resolution is not significantly affected by the small changeof δ. This small change (δ) in the spacing Δ, however, changes theoverall phase of the shearing interference pattern produced by the twogratings 140 and 150. In data acquisition, the spacing Δ is adjusted tohave different small shifts (δ) to obtain different shearinginterference patterns with different phase shifts for further processingwith phase extraction and unwrapping algorithms.

In addition, the specimen surface 130 may be tilted at different smallangles to produce different phase shifts in the correspondinginterference patterns in the CGS system. These and other phase shiftingmechanisms may be combined to effect desired phase shifts.

A CGS system may be designed with dynamically configurable shearingdistances to allow for obtaining data with different shearing distancesduring the measurement process as described above for improving theresolution and accuracy of the measurements. At least one of the twogratings in a CGS system such as examples shown in FIGS. 3 and 4 may beengaged to a positioning stage or positioning transducer to change therelative spacing between the two gratings in a controlled manner toachieve measurements at different shearing distances.

In addition to CGS systems, other shearing interferometer configurationsmay also be used to implement the shearing device 101 in FIG. 1. Severalexamples are provided below. These shearing interferometers use opticalelements different from gratings in CGS to generate the lateral shearingbetween two interfering wavefronts and have their respective uniquefeatures in measuring surfaces.

FIGS. 6A and 6B show two examples of cyclic shearing interferometers. Aparallel plate with one semi-reflecting surface is used to split thereceived probe beam into two beams. FIG. 6A uses a rotating transparentplate in one of the two beams to produce the shearing and the variableshearing distance. FIG. 6B uses a movable mirror in the optical path toproduce the shearing and the variable shearing distance by moving themirror away from a position with a zero shear. The phase shifting may beachieved by slightly translating one of the two reflectors, or bytilting the parallel plate with the semi-reflecting surface. FIGS. 7A,7B, and 7C show examples of Jamin shearing interferometers. FIG. 8 showsa Mach-Zehnder shearing interferometer. FIGS. 9 shows a Michelsonshearing interferometer. FIGS. 10A and 10B show two examples of parallelplate shearing interferometers capable of producing large shearingdistances. FIGS. 11A, 11B, and 11C show prism shearing interferometershaving two different prisms to produce the desired shearing. Structuresand operations of these shearing interferometers are well known. Theshearing distance may be controlled and adjusted by rotating a selectedoptical element in these interferometers. In general, the phase shiftingcan be achieved by tilting the specimen surface under measurement. Insome of these interferometers, one optical element in the optical pathmay be translated to produce the desired phase shifting without tiltingthe specimen surface. In any of these interferometer configurations, thespacing between the gratings can also be altered by causing therefractive index of the medium between the gratings to change in aprescribed manner. Furthermore, the grating itself can be effected byother means such as causing surface acoustic waves to propagate along asurface in the beam path, or by use of electrically-addressable liquidcrystal elements, etc. A means of effectuating a diffractive element maybe used in a CGS system.

In these non-CGS shearing systems, the uniformity of shearing distanceacross the field of view is relatively easy to control in comparisonwith a CGS system which needs to maintain parallel gratings as theseparation between two gratings is changed. These systems are alsorelatively easy to achieve smaller shearing distances by nominallysetting the systems at configuration with a zero shearing and by usingslight rotations to achieve small shearing distances. In addition, thesesystems avoid the use of precision holding mechanism for the gratingsand the in-line spatial filtering of unwanted diffraction orders in CGS.Due to these and other features, these non-CGS shearing interferometersmay be used to optically measure surfaces in certain applications whereCGS may be more difficult to implement.

The above non-CGS optical shearing interferometry systems may beconfigured and operated to achieve small shearing distances than the CGSdue to their designs. However, both CGS and these non-CGS systems may beoperated to achieve small effective shearing distances less than minimumshearing distances due to limitations of the mechanisms for adjustingthe shearing distances. For example, a shearing interferometer may beoperated to make two or more measurements at shearing distances withincremental differences. Two of such measurements may be combined toproduce an effective shearing distance at the difference between the twoclose shearing distances. Hence, this use of multiple shearing distancesenables highly accurate calculation of the estimated surface topologyfrom the relative data by a geometric calculation without using astandard numerical integration algorithm to compute the actual surfaceprofile. Details of this technique are provided below.

The characterization of high spatial frequencies (or low spatialwavelengths) in a shearing interferometer can be limited by the minimumachievable shearing distance, the minimum spot size of the measurementprobe (e.g. the pixel size of an imaging array), or a combination ofboth. In some shearing interferometers, the shearing distance may be themain limiting factor (shearing distance on the order of a fewmillimeters, pixel size on the order of 100's of micrometers or less)with the critical spatial wavelength corresponding to approximatelytwice of the shearing distance. Shorter shearing distances may beimplemented, but may result in a less sensitive interferometer. In areflection-mode shearing interferometers, for example, the slope perfringe=λ/2ω, where λ is the probe wavelength and ω is shearing distance.

If the configuration of a given shearing interferometer allowsadjustment of the shearing distance, multiple sets of interferograms maybe collected from the same specimen at difference shearing distances. Inthis case, when the data sets are taken in pairs, the effective shearingdistance of the two sets of data can be made to be the differencebetween the shearing distances of the two sets.

First, consider two measurements for a data set with two differentshearing distances ω1 and ω2, respectively, with the followinginterferograms:S(x ₁+ω₁ , x ₂)−S(x ₁ , x ₂)=n ₁λS(x ₁+ω₂ , x ₂)−S(x ₁ , x ₂)=n ₂λwhere n1 and n2 are represent the fringe orders where constructiveinterference occurs at n=0, 1, 2, 3, etc. and destructive interferenceoccurs at n−0.5, 1.5, 2.5, etc. The difference of two measuredinterferograms can be written asS(x ₁+ω₁ , x ₂)−S(x ₁+ω₂ , x ₂)=(n ₁ −n ₂)λThe above equation can be re-written asS(x ₁+(ω₁−ω₂), x ₂)−S(x ₁ , x ₂)=(n ₁ −n ₂)λHence, the combination of the two data sets yields a data set or newinterferogram having an effective shearing distance represented by thedifference of the two shearing distances of the individual data sets.Using this feature, the spatial frequency response of the system may beoptimized by selecting an effective shearing distance equal to the spotsize of the probe.

The practical implementation of this methodology may be achievedusing 1) the interferometer system designed with two distinctinterferometer paths of different shearing distances, 2) theinterferometer system with a single interferometer path, whose shearingdistance can be adjusted to obtain different inteferograms withdifferent shearing distances. Configuration 1 has the advantage that thetwo data sets can be acquired simultaneously and that the two paths arefixed and hence it is easier to maintain uniform and repeatable shearingdistances in each path. Configuration 2 has the advantage that it hasfewer components and hence can be more compact and less expensive.

In the CGS interferometer, the shearing distance may be adjusted bychanging the grating separation, probe wavelength or grating pitch.

Adjustment of the grating separation in CGS may be achieved using anactuator as described above. As an example, in a system configured withgratings having a pitch of 25 micrometers and a probe wavelength of632.8 nm, the grating separation would have to be increased by ˜39microns for each micron increase in shearing distance. In order toachieve changes in shearing distance on the order of a few micrometers,a piezo-electric transducer (PZT) system may be appropriate, whereas aprecision motor-driven stage system may be more appropriate for changesin the grating separation that result in changes in the shearingdistance on the order of 10's or 100's of micrometers. In either case,some additional metrology (e.g. displacement transducers) may benecessary to assure that the change in grating separation (and henceshearing distance) is uniform. Such a system has the advantage that theadjustment of shearing distance is continuous and the disadvantage thatit may be difficult to change the grating separation uniformly tomaintain uniform shearing distance across the field-of-view.

The use of the probe wavelength to change the shearing distance may beimplemented by using either distinct sources that are shuttered or byusing a laser in which the wavelength can be tuned (e.g. an Ar-ion laseror a tunable laser such as Ti:sapphire laser). As an example, in asystem configured with gratings having a pitch of 25 micrometers and afixed grating separation, the changing the probe wavelength from 632.8nm to 514 nm would change the shearing distance by 35.64 micrometers.Implementations of such a system may have the advantage that the changein the shearing distance can be made uniformly. In theseimplementations, only discrete changes in the shearing distance arepossible (based on available source wavelength) and the optical systemof the interferometer can be designed to respond identically to the twoor more wavelengths (through design or adjustment).

When the pitch of the gratings is used to change the shearing distancein CGS, two pairs of transmissive gratings that have a fixed linepattern on a glass or similar substrate may be used in two independentinterferometer paths. The two pairs have two distinct sets of gratings,each having different pitch. Alternately, a line pattern for a gratingmay be generated electronically or optically in a manner that isadjustable. For example, an acoustic grating may be used to produce theadjustable grating pitch for changing the shearing distance.

The configuration of the shearing interferometer system for a givenapplication depends on the power spectral density (amplitude versusspatial frequency) of the component being tested. Specifically, theslope sensitivity, λ/2ω, can be selected to ensure that the amplitudecan be characterized with an acceptable signal to noise ratio and theshearing distances can be selected to ensure that the spatial frequencycan be characterized (adheres to Nyquist's sampling theorem). In thisway, the system can be optimized for a given type of sample. Thepractical implication of the optimization is that the sample can becharacterized with the minimum amount of data, which in turn facilitatesefficient computation and analysis as well as data storage.

Some practical limitations may exist in selecting both the slopesensitivity and shearing distances. For the slope sensitivity, thepractical limit may be the intensity level resolution of the imagingsystem and the probe wavelength. As an example of a first orderestimate, a CCD array with 10-bit resolution (1024 gray scales)theoretically can resolve 1/2048th of a fringe (intensity variation fromblack to white represents ½ a fringe). If the probe wavelength is 632.8nm the minimum difference in height that can be resolved across theshearing distance is ˜0.31 nm (see equation 1). In practice, it may notbe possible or feasible to access the full dynamic range of the imagesensor and noise sources may limit the signal that can be extractedreliably. Maximizing the dynamic range of the image sensor and/orminimizing the probe wavelength may be used to characterize smalleramplitudes.

The selection of the shearing distances (and hence spatial frequencyresponse) may be subject to the some limitations and trade-offs. First,the in-plane spatial wavelengths cannot be smaller than approximatelytwice the probe wavelength. Second, for an image array/sensor of fixedsize, the field-of-view decreases linearly with the spot/pixel size.Third, the selected shearing distances define a region around the edgeof the sample over which interference data is cannot be collected. Thus,the larger the individual shearing distances, the more limited the datacollection at the edge of the sample becomes.

The above CGS and other optical shearing interferometry systems may beused to measure slopes and curvatures of various features and componentsformed on a substrate either directly or indirectly. In the directmeasurement, the probe beam can be directly sent to the patterned topsurface of these devices to obtain the curvature information. Thesurface features and components and their surrounding areas on the topsurface may be smooth and optically reflective to be accuratelymeasured. For example, some completed integrated circuits have a toppassivation layer, usually made of a non-conductive dielectric material,over the circuit elements on the substrate to protect the underlyingcircuits. The surface of the passivation layer is in general smooth andis sufficiently reflective for this direct measurements.

In some situations, the above direct measurements based on reflectionfrom the patterned surface may be difficult to implement. For example,features and components formed on the front side of a substrate or theirsurrounding areas may not be optically reflective. In addition, theeffectiveness and accuracy of this direct measurement based onreflection from the patterned top surface may be adversely affected ifthe properties of the features and components and their surroundingareas other than their slopes and curvatures significantly contribute tothe wavefront distortion because the wavefront distortion under suchcircumstance is no longer an indicator of the global slopes andcurvatures of the area illuminated by optical probe beam. The featuresand components on the front side may distort the reflected wavefront dueto factors other than the global slopes and curvatures, such as thelocal height of a feature or component being different from itssurrounding areas. In these and other situations, the curvatures of thefeatures or components may be indirectly measured by inference from thecurvature measurements of the corresponding locations on the opposite,unpatterned surface on the back side of the substrate. This is possiblebecause the stresses in the non-continuous features and componentsformed on the substrate can cause the substrate to deform and the thinfilms formed over the substrate generally conform to the globalcurvature of the substrate surface.

When the heights of certain features are different from theirsurroundings, the phase distortion on the wavefront of the reflectedprobe beam for each feature includes at least the portion contributedfrom the height difference and the portion contributed from thecurvatures. Since the backside surface is not patterned, any opticalinterferometer, including non-shearing interferometers may be used toprocess the reflection from the backside surface to obtain the surfacecurvature information. For example, non-shearing Twyman-Green andMichaelson interferometers may be used to obtain optical measurements onthe unpatterned backside surface of a wafer.

Notably, the patterned front or top surface of a wafer may be opticallymeasured with a phase-shifting shearing interferometer described aboveand the unpatterned backside surface may be optically measured with anyinterferometer including shearing or a non-shearing interferometer. Bothmeasurements may be processed or correlated to improve the overallmeasurements of the patterned front surface. The surface informationfrom the unpatterned backside surface may be used to provide the overallglobal surface slope information of the wafer. The surface informationfrom the patterned front side surface, which may be advantageouslyobtained from a shearing interferometer, may be used to provide detailedlocal surface information on the patterned front surface.

In implementation, the backside surface of a wafer may be supported bywafer supports in part because the patterned front surface, such ascircuits and other micro structures, may be damaged by contact of suchsupport members. FIG. 12 illustrates an exemplary layout for opticallymeasuring the backside surface of a wafer. The wafer supports in contactwith the backside surface may affect the optical measurements for beingpresent in the illuminate area and thus partially prevent the reflectedbeam to obtain the surface information in the areas occupied by thewafer supports. Such effects of the supports are undesirable and shouldbe removed.

FIG. 13 illustrates an example where the backside of the wafer issupported by three wafer supports that are oriented in a non-symmetricway to enable direct collection of data on the full wafer surface bymaking multiple measurements of the wafer at different angularorientations. Not shown is the hardware the places the wafer onto thethin supports in one of any number of angular orientations (placementand rotational devices like these are common in the automationindustry). FIG. 13 further illustrates that measurement of the backsideof the wafer results in an interference pattern that containsdiscontinuities because of the presence of the three point supports inthe measurement field. In a traditional arrangement, these fringes wouldprevent the conversion of the fringe pattern to meaningful data. Anumber of techniques are described here to allow measurements in theareas occupied by the wafer supports.

In one implementation, an interpolation algorithm is used to effectivelyinterpolate the fringe pattern across the discontinuities of the fringesdue to presence of the wafer supports. The interpolated fringe edgesenable the calculation of an inferred fringe that can be used in astandard interferometric processing algorithm. The algorithms used togenerate these inferred fringes may use a linear interpolation, a Splineinterpolation, a higher order polynomial interpolation, and a number ofalgorithms using spatial filtering and one of more of the previousdescribed techniques. The spatial filtering coefficients can be derivedby analyzing experimental and theoretical data on wafer deformationscaused by semiconductor and MEMs manufacturing processes.

Once the interpolation is completed, the software that drives the devicealso performs a ‘sense check’ on the resulting, inferred fringes basedon spatial frequency content and consistency with other fringes on thewafers.

In many cases, these algorithms will be sufficient to enable thecalculation of the processing of the interferometric fringe data intomeaningful information on wafer shape, slopes, curvatures, and stresses.However, in cases where higher levels of measurement resolution arerequired, the device will make measurements of the backside of the waferat multiple angular orientations. The device then compares the multipleimages for consistency and fills in missing data from one image (i.e.,parts of the wafer that were covered by the supports) with data fromanother image (i.e., an image that was acquired at a differentorientation, where a given part of the wafer covered in the former imageby the support pins, is no longer covered). The algorithms forperforming this calculation are straightforward.

The device may also use transparent, lens quality support pins that areessentially invisible to the probe wave front. These support arms andpins are machined from machine quality quartz and polished via a complexlapping process.

While this specification contains many specifics, these should not beconstrued as limitations on the scope of the invention or of what may beclaimed, but rather as descriptions of features specific to particularembodiments of the invention. Certain features that are described inthis specification in the context of separate embodiments can also beimplemented in combination in a single embodiment. Conversely, variousfeatures that are described in the context of a single embodiment canalso be implemented in multiple embodiments separately or in anysuitable subcombination. Moreover, although features may be describedabove as acting in certain combinations and even initially claimed assuch, one or more features from a claimed combination can in some casesbe excised from the combination, and the claimed combination may bedirected to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understand as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the embodiments described above should not be understoodas requiring such separation in all embodiments, and it should beunderstood that the described program components and systems cangenerally be integrated together in a single software product orpackaged into multiple software products.

Only a few implementations are described. However, it is understood thatvarious variations, enhancements, and other implementations may be made.

1. A computer-implemented method for determining stresses at a locationon a layered structure comprising at least one film formed on asubstrate, comprising: applying a spatially varying structural conditionin the layered structure to computation of stresses at a selectedlocation in the layered structure from curvatures at all locations ofthe layered structure, wherein the spatially varying structuralcondition in the layered structure comprises at least one of (1) thefilm covers only a portion of the substrate, (2) the film has a filmthickness that varies from one location to another across the film, and(3) the substrate has a substrate thickness that varies from onelocation to another; determining in the computation under the spatiallyvarying structural condition a local contribution to stresses at theselected location on the layered structure from curvature changes at theselected location; determining a non-local contribution to the stressesat the selected location from curvature changes at all locations of thelayered structure; and combining the local contribution and thenon-local contribution to estimate the total stresses at the selectedlocation.
 2. The method as in claim 1, further comprising: when the filmcovers only a portion of the substrate, applying a formula in thecomputation for an equivalent structure, which is identical to thelayered structure except for that the film in the equivalent structurefully covers the substrate, to compute the total stresses at theselected location.
 3. The method as in claim 2, further comprising:using a radial gradient of a sum of curvature changes along a radialdirection and an orthogonal circumferential direction to represent aninterface shear stress between the film and the substrate.
 4. The methodas in claim 1, further comprising: applying a formula to compute adifference in stresses along a radial direction and an orthogonalcircumferential direction that is independent of a thickness of the filmwhen the film thickness varies from one location to another across thefilm.
 5. The method as in claim 4, further comprising: using a radialgradient of a sum of curvature changes along the radial direction andthe orthogonal circumferential direction to represent an interface shearstress between the film and the substrate.
 6. The method as in claim 4,further comprising: using a sum of (1) a radial gradient of a sum oflocal curvature changes along the radial direction and the orthogonalcircumferential direction and (2) a non-local contribution to representan interface shear stress between the film and the substrate.
 7. Themethod as in claim 4, further comprising: using a formula to compute asum of local curvature changes at a given location along the radialdirection and the orthogonal circumferential direction wherein theformula is inversely proportional to a local film thickness at the givenlocation.
 8. The method as in claim 1, further comprising: obtaining afull field spatial curvature measurement of the layered structure; andusing spatial curvature changes from the full field spatial curvaturemeasurement to compute the local contribution and the non-localcontribution from distribution.
 9. The method as in claim 8, furthercomprising: obtaining diagonal curvature tensor components at theselected location from the spatial curvature change distribution;computing a sum of the diagonal curvature change tensor components and adifference of the diagonal curvature change tensor components, at theselected location; computing the local contribution to a sum of diagonalstress tensors at the selected location from the sum of the diagonalcurvature change tensor components at the selected location; computingthe non-local contribution to the sum of diagonal stress tensors at theselected location and a sum of diagonal stress tensors averaged over alllocations across the layered structure; and computing the localcontribution to a difference of diagonal stress tensor components of thelayered structure at the selected location from the difference of thediagonal curvature change tensor components at the selected location.10. A method for monitoring a substrate fabrication process using thecomputer-implemented method in claim 1, comprising: while the layeredstructure is being processed, directing an optical probe to the layeredstructure to optically obtain a full-field curvature map of the layeredstructure; processing the full-field curvature map to obtain curvatureinformation at all locations of the layered structure; applying thecomputer-implemented method in claim 1 to determine the total stressesat each location of the layered structure; and determining whether thelayered structure is defective based on an acceptable threshold stress.11. The method as in claim 10, further comprising using the totalstresses of the layered structure to determine whether a film is likelyto delaminate from the substrate.
 12. The method as in claim 10, furthercomprising using an optical shearing interferometer to optically obtainthe full-field curvature map of the layered structure.
 13. The method asin claim 12, wherein the optical shearing interferometer comprises acoherent gradient sensing system with two optical diffraction elements.14. A device for charactering stresses in a layered structure,comprising: an optical module to project an optical probe beam to alayered structure and to obtain a full-field curvature map of a surfaceon the layered structure; a processor in communication with the opticalmodule to receive data of the full-field curvature map, the processorcomprising: means for applying a spatially varying structural conditionin the layered structure to computation of stresses at a selectedlocation in the layered structure from curvatures at all locations ofthe layered structure, wherein the spatially varying structuralcondition in the layered structure comprises at least one of (1) thefilm covers only a portion of the substrate, (2) the film has a filmthickness that varies from one location to another across the film, and(3) the substrate has a substrate thickness that varies from onelocation to another; means for determining in the computation under thespatially varying structural condition a local contribution to stressesat the selected location on the layered structure from curvature changesat the selected location; means for determining a non-local contributionto the stresses at the selected location from curvature changes at alllocations of the layered structure; and means for combining the localcontribution and the non-local contribution to estimate the totalstresses at the selected location.
 15. The device as in claim 14,wherein the optical module comprises: a collimated radiation source toproduce the probe beam onto the surface of the layered structure; anoptical shearing interferometer device positioned to receive the opticalprobe beam reflected from the surface and to cause an opticalinterference between a reflected wavefront of the optical probe beam andanother replica of the reflected wavefront that is spatially shifted bya shearing distance, wherein the optical shearing interferometer isoperable to adjust a phase shift between the reflected wavefront and thereplica of the reflected wavefront to obtain a plurality ofphase-shifted interference patterns of different phase shifts; animaging device which captures the interference patterns to produce thefull-field curvature map.
 16. The device as in claim 15, wherein theoptical shearing interferometer comprises a coherent gradient sensing(CGS) system with two diffraction gratings.